HIGHER  GEOMETRY 

L I  VJfJL  11~/1  V   \JTJLrf  V/ iTl.  JL^  JL  JLV.1, 


WOODS 


University    of    California 

IRVINE 


/ 


THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

IRVINE 


IN  MEMORY  OF 

Dr.  Milton  Rosenthal 


HIGHER  GEOMETRY 


AN    INTRODUCTION   TO   ADVANCED  METHODS 
IN    ANALYTIC  (IEOMETRY 


BY 


FIIKDEUICK   S.  WOODS 

l-Kol   |.>sol;    HI-'    MATHEMATICS    IN    Till:    M  A  SS  Acll  [    -.KITS 
I.NMTTITL    OF     1  I'.CHNoI.oi;  V 


GIXX   AM)  (  OMI'AXY 


JE  li  f    3  t  lu  ii  .T  i'_"'_JP  r ' 8  a 

i, I  \N     \\1.    i  "V.  l'\NV  •   rKK- 

ri.ii  i .  ii.  ,  •  i  •  •  •  i  •  'N  •  i    -->.A. 


PREFACE 

The  present  Ixiok  is  the  outgrowth  of  lectures  given  at  various 
times  to  students  of  the  later  undergraduate  and  earlier  graduate 
years.  It  anus  to  present  soinr  ol  the  general  concepts  and  methods 
which  are  necessarv  tor  advanced  work  in  algebraic  geoiuetr\  (as 
distinguished  tViiin  dit'fei'eiit  ial  income tr\  ),  hut  which  are  not  now 
aeeessihle  to  the  Student  ill  ailV  olie  \i'lllllle.  and  thus  to  bridge 

the  '4'ap  between   the  usual  text    in   analytic  geometry  and  treatises 
or   articles   on    special    topics. 

With  this  object  in  view  the  author  has  assumed  vrrv  little 
mathematical  preparation  on  the  part  of  the  student  hevond  that 
acquired  in  elementary  courses  in  calculus  and  plane  analytic  <_:'eoni- 
etrv.  In  addition  it  has  hcen  necessarv  to  assume  a  slight  knowl- 
edge of  determinants,  especially  as  applied  to  the  solution  of  linear 
ei|Uatii>ns.  such  as  mav  he  acquired  in  a  verv  short  course  on  the  sub- 
ject.  Hut  it  has  not  heeii  assumed  that  the  student  has  had  a  course 
in  higher  algebra,  including  mat  rices,  linear  su  list  it  tit  ions,  invariants, 
and  similar  topics,  and  no  ettort  has  heeii  made  to  include  a  dis- 
cussion of  these  sllhjeets  ill  the  text.  This  restriction  ill  the  tools 

to  he  used  necessitates  at  times  modes  of  expression  and  methods 
of  proof  which  are  a  little  cumbersome,  but  the  appeal  t<>  a  larger 
number  of  readers  seems  to  justify  the  occasional  lack  of  elegance. 
In  preparing  the  text  one  of  the  greatest  problems  has  consisted 
in  determining  what  matters  to  exclude.  It  is  obvious  that  an 
introduction  to  geometry  cannot  contain  all  that  is  known  on  anv 
subject  or  even  refer  bricllv  to  all  general  topic-.  The  matter  of 
selection  is  necessarily  one  of  individual  judgment.  <  >ne  lapjv 
domain  ot  u'coniet  rv  has  been  delililtelv  excluded  troin  the  plan  ol 
the  book  :  namely,  that  of  differential  geometry.  In  the  tield  uhieh 
i-  left  the  author  cannot  dare  to  hope  thai  his  choice  of  material 
will  agree  exactly  \\ith  that  which  \\oii  Id  he  made  by  an\  other 
teacher.  lie  hope-.  howes  er,  that  his  choice  has  heel]  sufficiently 
\\i-e  lo  make  I  he  hook  USelul  |o  liial!\  besides  Ililllscll 


iv  PRKFACK 

The  plan  of  the  hook  calls  for  a  study  of  different  coordinate 
systems,  hased  upon  various  geometric  elements  and  classified 
according  to  the  inuiiher  of  dimensions  involved.  This  leads  natu- 
rallv  to  a  final  discussion  of  /^-dimensional  geometry  in  an  ahstraet 
sense,  of  which  the  particular  geometries  studied  earlier  form  con- 
crete illustrations.  As  each  system  of  coordinates  is  introduced,  the 
meaning  of  the  linear  and  the  quadratic  equations  is  studied.  The 
student  is  thus  primarily  drilled  in  the  interpretation  of  equations, 
hut  acquires  at  the  same  time  a  knowledge  of  useful  geometric  facts. 
The  principle  of  duality  is  constantly  in  view,  and  the  nature  of 
imaginary  elements  and  the  conventional  character  of  the  locus  at 
infinity,  dependent  upon  the  type  of  coordinates  used,  are  carefully 
explained. 

Numerous  exercises  for  the  student  have  been  introduced.  In 
some  cases  these  carry  a  little  farther  the  discussion  of  the  text, 
hut  care  has  been  taken  to  keep  their  difficulty  within  the  range 

of  the  student's  ability. 

FRKDKKK'K  S.  WOODS 


CONTEXTS 

PART   I.    (1KNKKAL  CONCKPTS  AND  (  )XK-I)IM  KXSION  AL 

(JEOMETRY 

CIIA1TKK    I.    (iKNKRAL  CONCKITS 


1.  CiMirdinates 

•_'.  The  principle  '>f  duality 

'•\.  'I'll'-  u-e  iif  ima^inaries 

I.  Ititinitv 

.").  'I'ranst'uriiiatioiis 

»!.  (  iroups 


niAiTKK  ii.   KA\(;KS  AND  TKNCILS 


7.  Cartesian  coordinatr  nf  a  point  on  a  lin 

•s.  I'rojcctivc  I'liiinlinati'  of  a  point  on  a  li 

'.i.  Change  nt'  coordinates 

in.  Coi'irdinate  of  a  line  of  a  jiencil 

11.  Coonlinatc  of  a  iilanc  of  a  iiencil  . 


!'_'.  The  linear  transfoniiat ion    . 

1:1.    The  cross  rat  io 

11.     1 larnionic  sets     

l.'i.     I'rojection 

It).  l'er<]ieet  ive    li'4'lires 

17.  <Hher  one-<liliiensional  exten 


1'AKT    II.    TWO  -DIM  KNSION  A  I.  (JKOMKTHY 

(   HAITKK    IV.     I'oINT    AND    LINK    < '<  ><  >K  Dl  N  ATKS    IN    A    I'LANK 

Is.  Homogeneous  ('artesian  pnint  coordinates 

1  (.i.  The  •,!  ra'r_;ht    line 

•_'M.  The  cirele  po'int<  at    intinity 

L'l.  The  conic 

'J'J.  'I'rilinear  point  cm  >rdinates 

-'•>.  l'i  lints  on  a  line 


vi  (ONTKNTS 

SI  i  TIi 'N  I'AfiK 

•_'!.    The  linear  equation  in  ]i«iint  coordinates.      .           ;jtj 

L'.'i.     Lines  of  ;t  pencil ;;7 

'Ji'>.    Line  coordinates  in  a  plain' :is 

'J7.  IVii'-il  i«f  lines  ami  the  liip-ar  equation  in  line  coordinates        .      .      .  :!!i 

•J-v     Dualistie  relations ll> 

•_".'.      (  'liaise    ,,f    coordinates 11 

:!n.    Certain  strai^'lit-Hne  eontiijurations 11 

ol.    Cnrve>  in  jHiint  CM n'inli nates ."id 

;!'_'.    ('iii'\c.--  in  line  coordinates ~>:\ 

(  HAl'THi;    V.    Ct   KN'F.S   (>F   SKCOM)   (tlll)Ki;    AM)   SKCo.N.I)    CLASS 

>'.].     Sin^nlai1  I'dints  n!  a  enrve  of  seeonil  order ."iS 

i  1.  1'nles  and  |>olars  \\itli  respect   to  a  curve  of  .second  order      ....  "ill 

l.'i.    Classification  of  ciir\es  of  second  onler i;.") 

111.    Singular  lino  of  a  curve  of  second  class Ii7 

17.    Classification  of  curves  of  second  da>s us 

;--.  I'olo  and  polars  \\itli  respect  to  a  curve  of  second  class       ....  70 

ill.     I'rojective  properties  of  conies 7'2 

ril.MTKi;    VI.    Ll.NKAi;    TIJANSFOKMATIUNS 

In.    Collitieations 7s 

11.    Types  nf  noiisin^nlar  collineations N:; 

!'_'.    Correlations ss 

1:1.     Pairs  of  conies H.'i 

1  1.    The  protective  -i-oiip inn 

l.'i.    The  metrical  -'roiip Inl 

1'i.     Alible  and  tiie  circle  points  at   intinitv In.") 

MIAlTF.i;    VII.     I'Kd.IKCTIVK    M  F.  AST  K  KM  F.  NT 


CIIAI'TKK    VIII.    CONTACT   TIJ  A  NSF<  >I!  M  ATI<  >NS    IN    Till-:   I'l.ANK 


CONTKNTS  vii 

CIIA1TKK    IX.    TKTKACYCI.1CAK   ( '<  M  >K  I  )I  N  ATKS 

SKrTIoN  I1  U.K 

f>7.    Special  tetracyclica!  ciMirdinates M^ 

.">>.    Distance  lietueen  two  [mint-. M'.i 

.V.I.    Tlir  circle 1  ID 

tin.     lielation  between  tctracyclical  and  Cartesian  cufinlinates    .      .      .      .  1 1'J 

til.    Ortho-onal  circles Ill 

ti'J.     Pencils  of  circles 1  M 

<i:;,    'I'he  general  tetraeydieal  coordinates \~>n 

til.    Orthondiial  ciionliiiates I.',:; 

(i."i.    The  linear  transfonnatiuii 1  •")  1 

liti.     The  metrical  tniiisfoniiatinii \'<~i 

ti".     IllVel'siull 1'i'i 

fi^.    The  lineaf  L;n>ii|> l.">',( 

till.     Duals  of  tetracyclical  coiinlinates Ml 

CHAl'TKK    X.    A    Sl'KCIAK   SVSTK.M    t»K   Ct )(")!{ DI N  ATKS 

TD.    The  ciHinlinate  system Ml 

71.    The  straight   line  ami  the  equilateral  hy|ierl><>la M'! 

7'J.    '1'lie  1'ilinear  eipiat  ion Iii7 

7o.    '1  he  bilinear  transformation    .                                                                        .      .  M'.i 


PAKT    III.    TllKKKDl.MKNSIOXAL  (JKOMKTKV 

CHAl'TKK    XI.    CIKCI.K    C( »( >K  DI  N  ATKS 

71.    Elementary  circle  coi'inliiiatcs 

7-i.     I  lie  quadratic  circle  cumplex 

7'i.     Higher  circle  corirdinates 


CHAl'TKK    XII.     1'OINT    AND    I'LANK    C<  '<  >K  DI  N  ATKS 


77.  ( 'artesia  n  ]  >oi  nt  cm  .i-ditiates      .... 

7--.  Distance 

7'.'.  The  >t  rai^ht   line 

MI.  The  plane 

>  1 .  1  •  ire  c  1 1  ( 1 1 1  a  n '  1  a  n  u 1 e 

v'_'.  C^uadriplatiai1  point   coc'irdi nates    . 

>M.  St  rai-  lit   line  and   |  >la  ne 

s  I .  I  'lane  coordinates 

>••">.  One-dimensional  extent-,  of  points    . 

>ii.  LIICIIS  of  an  equation   in  point  cooi'dinat 

>7.  <  )ne-i  ii  ineii^ional  I'Xtenls  of  plane-    . 

s^.  LIICMS  ii|    an   equation    in    plain-  coordinat 

"•''•I.  ( 'halite  nt   CIM  i|-i  Ij  nate-    . 


viii  CONTENTS 

CHAriF.K    XIII.    SIRFACKS   oF   SF.CONI)    OKDF.K    AND   OF 
SKI '(  >N  D   I  LASS 

SKCTION  I'ACK 

!K).      Surfaces   of   see, ,nd    order •_'•_'() 

'.il.    Singular  pn'mts •_>•_>! 

'.r_>.     1',,1,-x  and  pnlars •_>•_>•_> 

!•:;.    Clarification  nf  surfaces  of  second  order •_'•_'! 

i'l.    Surfaces  nf  secnnd  ni'der  in  Cartesian  cnnrdinates '2'27 

!'."i.    Surfaces  nf  second  order  referred  t<>  rectangular  axes •_'•_'!! 

!'i!.     Killings  on  surfaces  of  second  order      .      .  •_!:'>•_' 


ciiArri;i;  xiv.  TKANSFOUM. \TIONS 


ion.  ('ollincatioiis 

lnl.  Tyi'i's  nt"  iioiisin^uliir  ciillincat  inns 

llfj.  Cnnvlatiniis 

lu:!.  The  projcrtivc  and  the  metrical  turnups. 

]u|.  I'rojrctivc  ^ciuiict  rv  nn  a  ((iiadric  surface 

!().").  I'l'i  iject  i\  e    measurement 

lnii.  ClilV,,rd  parallels 

In7.  Cmitact  ti'ansfnrmat 'K ins 

ln>.  I'nint-pdiiit   traiisfnrmatiniis        .... 

lii'.i.  I'oint-sHi'face  tratisfurmatioiis    .... 

lid.  I'oint-cnrve  transfnrmat  inns 


CIIAI'TF.K    XV.    TIIK    Sl'H  Kli  K    IN    CAKTF.SIAN    ('( H  JUDIXATES 

111.  I'ellciN   ,-f    S].lieres 

11-J.  linn. lies  ,,f  spheres         

I  I:1,.  Complexes   (if    sphere- 

I 1  }.  Inver.Miiii 

li:..  Dllpin'-   eyclide 

lit'..  ('Vclide.s 


CIIA1TKIJ    XVI.     rFNTASl'HF.KK'AL  COOIM)! NATKS 

1 17.    S]«'cia!i/ed  cnnrdinates 

1    K       The     ,p|,,    re 

1  1  !i.     A  ir_;le  Let  u  c.-n  spheres 

1-".      The    pnwer   nf   a    pi.inl    \\  ith    rex]K'Ct    In   ;i   spliiTI' 

1'Jl .    I  rVjieral  i  <r\  In  r'<  MI.I!  cnnrdinates 


CON TK NTS 


IX 


SK<  TIdN  I'Vi.K 

1'J'J.    The  linear  transformation -_".jl 

1'Jo.     Relat  i(  'ii  ln-t  \vi-i-u  peiitasplierical  ami  Carte-iaii  cm  irdinate^         .      .  '2'.>'-\ 

ll'l.     iVncils,  I'liiidles,  and  complexes  of  spheres -_".i:; 

l'_'">.    Tangent  circles  and  spin-res L".'~> 

1  •_'»!.    Cyclic!. •-,  in  penta-pheriral  codt-dinates I".i7 


1'AKT    IV.    OKOMKTKY   OF   FoCR    AND    IIKIHFK 
DIMENSIONS 

CIIAl'TKi;    XVII.     LINK    O  ><")!{  I  >I  NATKS    IN   THIiKI-: 
DIMKNSIMNAI.   STACK 


1-J7. 
l'_'s. 

1  L".l. 

l:in. 
lol. 
!:'.'_'. 


The  I'luck.-r  (•(.(irdinates 
I  )uali>t  ic  iletinit  imi 

1  tlter-ect  ill'^'    lilies 

(  ieiieral  line  cc  H  ipl'mates 
1'eiicils  ami  luindles  i>|'  li 
Complfxes,  ccn^ruences. 
Tin-  linear  line  complex 


Cdinplexes  iii  pdinl  coordinates      .      .      .      . 

Cdinplexes  in  ('artesian  coordinate's   . 
The  liilinear  ei|iiatiiui  in  pdint  coi'irdi  nates 
The  linear  line  cdn^riience    ...... 

The  cylindrdid     .......... 

The  linear  line  >eries    ........ 

The  .  jiiadrat  ic  line  cdinplex         ..... 

Singular  surface  n\  the  .piadratic  cdmp]r\ 
1'liicker's  Cdinplex   >nrt'aces     ...... 

Till'    ('_'.   -J)    Cdimnielice          ....... 

Line  ci  in^nience.>  in  ^'en.-ral 


CHAI'TKi;    XVIII.    M'lIKKI.   ( '<  H  >|;  J)I  N  ATKS 


ll'i.  Kletiientarv  sphere  cdnrdinates 

117.  II  iuher  >ph>-re  ci  Hirdinates     .      .      .      . 

1  I  ^.      A  liule    liet  Ween    Spheres 

1  1'.'.  The  linear  cdinplc\  df  dri.-nted  ^pheri 

I..D.  Linear  cdn^niem-e  dt  drienteil  -phei-c 

I'd.  Linear  series  dt'  oriented  spheres    . 

1'i'J.  I'eiiciK  and  1  iii  ii'll'---  i.|'  lan^enl   -ph.-r. 

l.i-l.  (Quadratic  complex  ,.f  i.rieni'-d  sjiliere 

l."il.  I)nalitv  of  line  ami  -pheiv  ^edinetry 


x  OiNTKNTS 

(ll.M'll.i;    XIX.    Full;    IH.MF.NSlnNAI.    I'(  )I  NT   (  '<  ><  )]{  I>1  N  A'I'F.S 


1  .").">.  I  >rtitlitioIlS 

l.'.i;.  [iiii-r-t-rtioiis 

l.">7.  Knclideaii  space  of  t'tiiir  dimensio 

i:.v  Parallelism 

l.V.i.  IVrpi'iitUriihirin 

1  i!i  i.  M  j  ilium  in   liiH--.  plini 

1'il.  IlypiT.-iirl'aci's  i.f  si'i' 

1'i'J.  l)iialiiv    lii'twiM-ii    lii 

','<•<  >iiiet  r\    ill    t'oin- 


niAiTKK  xx.  (,F.D.MI:TI;V  OF  A'  DIMF.NSIONS 

lii:l.  I'l-MJ.Tt  i\  i'  xpacc        .................  ;;ss 

Ml.  Intersection  of  linear  spaces       .............  :;'.H) 

I'i.'i.  The  c|iiadratic  hvpcrsnrt'act1         .............  :5!i'J 

1'i'i.  Inti-r.-ec'tion  of  a  ipiadric  l>y  li\  perplane-,     .....      ....  :i!iti 

l'!7.  Linear  >pace>  on  a  ipiadric    ..............  (ill 

Iti^.  ^(ereo^rapliic  projection  (lf  a  (|iiadric  in  \,  upon  Sn  _,       ....  |l)7 

li'i'.i.  A  I'plicat  ion  to  line  ^voniet  rv      .............  |ln 


IXDKX  li'l 


HIGHER  GEOMETRY 

PART   I.     (iKNKHAL  CONVKITS   AM) 
ON K -1 ) I M KNSLOXA L  ( i KO.M KTK V 

(TIAPTKR   I 

GENERAL  CONCEPTS 

1.  Coordinates.  A  set  of  //  variables,  the  values  of  which  lix  a 
geometric  ob|ect,  are  called  the  rtiorthnnt^x  ot  the  object.  1  he  ana- 
Iviic  geonietrv  \vhieh  is  developed  bv  the  use  of  these  coordinates 
has  as  its  </<'//;<•///  the  object  tixed  bv  the  coordinates.  The  reader 
is  familiar  with  the  use  of  coordinates  to  lix  a  point  either  in  the 
plane  or  iu  space.  The  point  is  the  element  of  eleineiitarv  aua- 
Ivtic  geometrv,  and  all  figures  are  studied  as  made  up  of  points. 
There  is,  however,  no  theoretical  objection  to  using  anv  geometric 
ti'_nire  as  the  element  ot  a  ^cornet  rv.  In  the  following  pac_;'es  we 
shall  discuss,  among  other  possibilities,  the  use  of  the  straight  line, 
the  plane,  the  circle,  and  the  sphere. 

The  tUi/irnxiunx  ot  a  svstein  ot  geomeirv  are  determined  bv  the 
number  of  the  coordinates  neeessarv  to  lix  the  element.  Thus 
the  geonietrv  in  which  the  element  is  either  the  point  in  the  plane 
or  the  straight  hue  in  the  plane  is  two-dimensional;  the  u'eoinetry 
in  which  the  element  is  the  point  in  space,  the  circle  in  the  plane, 
or  the  plane  in  >pace  is  three-dimensional:  the  '''comet  rv  in  which 

II  O  . 

the   element    is   the    straight     line   or   the    sphere    iii    space    is    four- 
dim  e  n  >  i  n  n  a  1 . 

Since  each  coordinate  mav  take  an  infinite  number  ot  values, 
the  fact  thai  a  geonietrv  has  //  dimensions  i<  often  indicated  hv 
saving  that  the  totalitv  of  element-,  form  an  f."  extent.  Tim-  the 
points  in  -pace  form  an  •/.'"'  extent,  while  the  sii'ai^hl  line-  in 
space  form  an  /  '  extent.  It  m  an  //'  extent  the  coordinates  ot  an 
element  are  connected  bv  /'  independent  condition-,  the  elements 

1 


2  <>NK -lUMKNSinXAL  (JKOMKTRY 

satisfying  the  conditions  form  an  -fJ'  '  extent  lyin^  in  the  x" 
extent.  Thus  a  single  equation  between  the  coordinates  ot  a  point 
in  space  defines  an  f.~  extent  (a  .surface)  Ivin^'  in  an  r.''  extent 
(space),  and  two  equations  between  the  coordinates  ot  a  point  in 
space  dctine  an  s. ''  extent  (a  carve). 

2.  The  principle  of  duality.    When  the  element  has  been  selected 
and    its  coordinates  determined,  the  development   ot    the  j^eometrv 
consists  in   studying  the  meaning  of  equations  and   relations  con- 
necting the  coordinates.     There  are  therefore  t\vo  distinct  parts  to 
anahtic  -jvonietrv,  the  analvtie  work  and  the  geometric  interpreta- 
tion.   Two  svstrins  of  <4Vonietrv  depending  upon  different  elements 
\\iih  the  same  nnmher  of  coordinates  will   have   the   same   analytic 
e\pre>sion  and  will  differ  only  in   the  interpretation  of  the  analy- 
sis.    In  such  a  case  it   is  often   snl'licieiit    to  know  the   meaning  of 
the  coordinates  and  the  interpretation  of  a  few  fundamental   rela- 
tions in   cadi   system  in   order  to  tind   for  a  theorem    in   one  ^eoin- 
etrv   a  corresponding  theorem    in    the  other.     Two  systems   which 
have   such  a   relation  to  each    other   are  said   to   he  iliiiilixt'n;   or  to 
correspond  to  each  other  l>v  the  nriitfiiilf  »t' ilun/if //. 

It    is  ohvionslv  inconvenient   to  inve  examples  of  this  principle 
at    this   time.  l»nt    the    reader   will   tind    numerous  examples   in    the 

pa'4'es     of     this     hook. 

3.  The  use  of   imaginaries.     Between    the  coiirdinates  of  a  geo- 
metric element  and  the  element  Itselt  there  tails  to  he  pel'lect    equiv- 
alence  unless  the  concept   ot   an   imaginary  element    is  introduced. 
('oiisider,  tor  example,   the  usual  ('artesian  coordinates  (r,  _// )  of  a 
point  in  a  plane.     If  we  understand  hv  a  "  real  poinl  "  one  \\hich  has 
a   po>ition   on  the  plane  wliieh   ma\'  he  represented   h\'  a  pencil  dot. 
then  to  anv  real  pair  ot   values  ot  .r  and   //  corresponds  a  real  point, 
and   ciiiiyrrsrly.     It   is  highly  inconvenient.  ho\\c\'cr.  to   limit   our- 

selves    111    the    aiialvlic    Wol'k    to    real     \allles    of    the    \ariahles.       \\*e 

accordiiiLi'U    introduce  the  convention   ot   an       imaginary  point      l>v 
s, IN  in-   iii.it    a   pair  of  values  of    ,   and  //  of  which  one  or  hot  h   is  a 
complex    i  plant  it  v  de  tines  siidi   a    point.      In    this  sense  a       point 
is  nothing  inure  than  a  concise  expression   tor      a  \aliie  pair  ( ./'.  // ). 
I*  rom     tins    standpoint     many    propositions    ot     analytic    ^cometr\ 
are   partly  theorems  and    parllv  ddinitions.     I-'or  example,  take  the 
pi    :  ciiiiat  ion  of  the  tirsl  decree  represents  a  straight 


(JKNKKAL  CONCEPTS  3 

line.  This  is  a  theorem  us  t'ur  us  ivul  points  und  reul  lines  ure 
concerned,  hut  it  is  u  detinition  tor  imaginary  points  suiist'\  in^  un 
(•(juution  \\'ith  real  coefficients  und  for  ull  points  satisfying  un  eipiu- 
tion  with  conipli-x  coeilieients.  The  detinition  in  <jiirstii>n  is  thut 
u  struight  line  is  the  totulity  of  ull  vulue  puirs  ( ./•,  // )  which  >utisfy 
uiiv  lineur  e([iiutioii. 

Anv  pmposit ion  proved  for  reul  1  inures  niuv  l»e  extended  to  ima'_r- 
inurv  ligmvs  pro\  ided  thut  the  proof  is  pnrelv  un  unuK'tie  one 
which  is  independent  of  the  reulitv  of  the  (juuntities  involved. 
One  cannot,  however,  extend  theorems  which  ure  not  unulvtie  in 
their  nutnre.  I-'or  example,  it  is  proved  for  u  reul  triangle  that  the 
length  ot  unv  side  is  less  than  the  sum  ot  the  lengths  ot  the 
other  two  sides.  The  length  of  the  side  connecting  the  vertices 
(.r},  //!  )  and  (./•.,,//._,)  is  \'7(  .'\ —.''.,)"+(//,—  //.,)'•  \V»'  may  extend 
this  detinition  ot  length  to  imuginury  points,  hut  the  theorem  con- 
cerning the  sides  of  u  triangle  cunnot  he  proved  unulvtieullv  and 
is  not  true  for  imuu'inaries,  us  niuv  he  seen  hv  testing  it  for  the 
triangle  whose  vertices  are  (<>,  <•).  (  ''.  1  ).  and  ( /.  —1  ). 

Similar  considerations  to  those  we  have  just  stated  for  u  point 
in  a  plane  apply  to  any  element.  It  is  usual  to  have  u  real  element 
represented  hv  real  coordinates,  hut  sometimes  it  is  found  con- 
venient to  represent  a  real  element  hy  complex  coordinates.  In 
either  ca>c  there  will  he  found  in  the  analvsis  certain  comhinat ions 
of  coordinates  which  cunnot  represent  reul  elements.  In  all  cases 
the  ^eometrv  is  extended  h\  ihe  convention  that  such  coordinates 
represent  imaginary  elements. 

4.  Infinity.  Inlinitv  may  occur  in  a  system  of  L;vonietrv  in  two 
wuvs  :  tirst,  the  vulue  ot  one  or  more  ot  the  coordinates  niuv  increase 
without  limit,  or  secondly,  the  element  which  we  suppose  1\  inu' 
within  the  ran^'e  of  action  ot  our  physical  senses  may  he  so  displaced 
that  its  distance  from  its  original  position  increases  without  limn. 

Intinitv  in  the  tir.M  sense  mav  he  avoided  h\  \\ntni'_;'  the  coi'ir- 
d  mutes  in  the  ton  n  ot  rut  ios.  1  or  a  rat  10  increases  wit  lion  t  limit  when 
its  denominator  approaches  /ero.  ( 'oi'ird  in  at  es  thus  written  ure  culled 
/iu//iiii/f-/(,i>iiK  I'niirili nut!-*,  hecunse  eipiutJons  \\iMlteii  in  them  heconie 
hoiuoj^eiieons.  I  hev  are  o|  constant  n>e  in  this  hook. 

'Ihe  treatment  ot  intinitv  in  the  >ecoiid  sense  i>  not  >o  Dimple. 
hut  proceeds  as  follow^  :  As  an  element  of  the  geometry  recedes 


.(  <>NK-1>IMKNSU)NAL   < !  KO.M  KTK  Y 

indefinitely  from  its  original  posh  ion,  its  coordinates  usually 
approach  certain  limiting  values,  which  arc  said  l>v  dctinit ion  to 
represent  an  clement  at  inlimtv.  '1  he  coordinates  ot  all  ele- 
ments at  infinity  usually  satisfy  a  ceilain  equation,  which  is  said 
to  represent  the  Incus  at  infinity.  '1  he  nature  ot  this  locus 
depends  upon  the  coordinate  svstem.  '1  hus,  in  the  plant1,  l>v  tin- 
use  n('  one  sNstem  ot  coordinates  all  "points  at  infinity"'  are  said 
tii  he  mi  a  straight  line  at  infinity  ;  hy  another  system  ot  coor- 
dinates the  plane  is  said  to  ha\e  '  a  single  real  point  at  infinity  : 
liv  still  another  system  of  coordinates  the  plane  is  said  to  have 
t  \\'o  lines  at  infinity."  These  various  statements  are  not  contra- 
dictory, since  they  are  not  intended  to  express  any  tact  about  the 
physical  properties  of  the  plane.  They  are  simply  conventions  to 
express  the  \\  av  in  whiih  the  coordinate  system  mav  l>e  applied 
to  inlinilelv  reunite  elements.  Then-  is  no  more  difficulty  in  pass- 
in  LT  trnin  one  convention  to  another  than  there  is  in  passing 
from  one  coordinate  system  to  another.  The  convention  as  to 
elements  at  infinity  stands  on  the  same  liasis  as  the  convention  as 
tn  imaginary  elements. 

5.  Transformations.  A  transformation  is  an  operation  by  which 
each  element  of  a  Ljvoinet  rv  is  replaced  1»\  another  element.  The 
ne\v  (dement  may  lie  ul  the  same  kind  as  the  original  element  or 
nt  a  different  kind.  For  example,  a  rotation  of  a  plane  ahont  a 
tixeil  pmnt  is  a  transformation  of  points  into  points;  on  the  other 
hand,  a  transformation  may  l»e  made  in  the  plane  l>v  which  each 
point  ot  the  plane  is  replaced  hv  its  polar  line  with  respect  to  a 
fixed  conic.  \\  c  shall  consider  in  this  hook  mainly  tnmli/tn-  //•<///*- 
t'lnmit n,iin,  that  is,  those  m  \\hich  the  coordinates  ot  the  trans- 
formed clement  are  analytic  functions  of  those  of  the  original 

element . 

A  transformation  mav  he  conveniently  expressed  l>v  a  single 
s\  mliol.  siich  as  '/'.  If  \\  e  wish  to  express  the  fact  that  an  element. 
or  a  ei  i!iti-_ni  rat  mil  o|  elements.//,  has  1  n-eii  t  ransl  drilled  into  allot  her 
'•lenient  or  coiil'iM-iirat  ion  /-.  \\'c  \\rite 

(  1  ) 


CKNKKAL  CoNCKl'TS  ."> 

result  is  a  single  Iniusformutinii  fi,  and  we  write 

f/  =  ,ST,  (-J) 

where  <!  is  called   the  i>r<«lu<-t  of  ,S'  and    '/'. 

Siniilurlv,  the  earning  out  in  succession  ot  the  transfoniiiitioii 
'/',  then  X,  and  then  //.  is  the  product  /,'>"/'.  This  svmbol  i>  to  be 
interpreted  as  meaning  that  the  transforinat  ions  ai'e  to  lie  carried 
out  in  order  from  ri'_dit  to  left.  This  is  important,  as  the  /(/•<»///<•/ 
'</  trttHxfunmttiunx  /x  H"t  mvr.s.xv//'////  <-i>//i////ifnt//-,'.  For  example,  let 
7'  he  the  moving  of  a  point  through  a  fixed  distance  in  a  fixed 
direction  and  S  the  replacing'  of  a  point  liv  its  svinniet  rical  point 
with  respect  to  a  lixed  plane.  It  is  evident  in  this  case  that 

X7'-7'.s'.  (:',) 

.1  jtrmlnt't  iif  t rii iixt'nriiiitf /'in*  /x,  /m/i'i'i'i  /•,  (txxwittth't'.  To  pro\c  this, 
let  /i',  .s',  and  7' be  three  t  ransl'oriuat  ions.  \Ve  wish  to  show  that 

( /;  N  )  T=  j;  (  s  T  )  =  y.'.s  r.  ( 4  j 

In  the  sense  of  formula  (  1  )  let 

T(n)=l,  ,S'(/»)=f,  /,'('•),=  ,/. 

Then  (  //,S')  7'(^  >: •--  i;S(l>)  —  J!(  r):    i/. 

On  the  other  hand.    ST(tt  )  =  S  (_/>)=  c, 
so  that  /,'(  N7'  )  (<i)  -----  //( (•)  =  </. 

This  establishes  the  t  heoivm. 

If  7'  represents  an  operation,  7'  '  shall  represent  the  hn'i-rxi' 
operation:  that  is,  if  7'  transforms  anv  element  <t  into  an  element 
I',  T  '  shall  transform  everv  element  /<  back  into  the  original  <i. 
The  product  then  of  '/'  and  7'  '  in  an\  order  leaves  all  elements 
unchanged.  It  is  natural  to  call  an  operation  which  leave.--  all  ele- 
ments unchanged  an  /</(•////<•'//  transformation  and  to  indicate  it  b\ 
the  s\  nibol  1.  \\'e  have  then  the  equation 

TT    '   -    7'    '/'--  1.  (o) 

II   X  and    7' an-  two  transformations,  the  operation 

7',s'7'   '      .s"  (ii) 

is  called   the   //•<///>•/'"/•///   of  .S'  b\     7'. 

If  Nj  and  .s'.'.  are  the  transforms  of  N,  and  N  I'espect  ivelv,  then 
N,'N'.  is  the  transform  of  .s',.s',.  l''or 

N.S",     (Ts,r  '(7'.s:.y  -)     y\  y  'y.s, 7    '     7'<  N,N  >  7'  '. 


t;  oNi:    DIM  KNSM  >NAL   <  I  K<  >.M  KTK  V 

EXERCISES 

1 .    St.itf  \\  h;di  "t  i  in'  following  pairs  of  operations  an-  commutative  : 

•  i    ,i  t  ran  slat  ion  and  a  rot  at  ion  about  a  fixed  point  ; 
i  /.  i    t  uo  n  'la!  ions  ; 
i  ,  i    t  \vo  1 1  an.slal  i>  '!!•>  : 

i  otal  i'  -M  and  a  relied  ion  on  a  line. 

•..'.    i;    x  t..  ;i  transformation  Midi  that   >>'J— 1.  prove  that   >'   '  -  >',  and 
,,,;.•,,;-..'.•,.    i  i ;  \ .    '_;ei  iinet  rie  I'xaiupli'S  <  >t  t  i'a  us  lor  mat  ions  oi  tins  t  v  pe. 

;;.      |'|,    ,,      •;,.;'     •[;,',.   l  |  .]•(  ..  -al   o  t    t  !  If   pP  M  1  U(  't   of   t  \Vi  •  t  la  1 1>  fol'll  Kit  iollS   is 

al>  of  the  t ran>fonuat ions  in  inverse  order; 

•  '  .     |  lllLlt    .   /,'>    /     "      '  '/'     '>      '/.'      '. 

-j.    I  •    s   i-   a    rot  at  ion   in  a   plane  and   7'  a   translation,  liml  t  he  trans- 

:     x   IP\     I     ind  t  he  I  ranstorin  of    '/'  1  >\    N. 
;,.    I'm       thai   the  transform  of  the  inverse  of  >'  is  the  inverse  of  the 

:  of   Y 

»;.    h   the  pro.p.ii-t  "t   two  transformations  i>  eommiitat ive.  show  that 

i  i'A  !:     t  rallsfi  'I'lll     liV    t  he    '  't  hel'. 

6.  Groups.    .1  >••  /  "i'  friin!<t'"i'/iiiili"iix  /•'/•///  <i  ///•""/'  /'  //"  *''t  <'"ii>it/i<x 

'/,,     ,'///•,  /-.v,        t'  ,,;/•>/  f/-itlixt'"t'/inlfi"lt    "t'  tli,     .v,7    ilii'l   /'    tin    jii'mlni-t  "t'linil 
'r  ii  -~  i'      14 a   ->'  '/i'    mt    /x   "/>•"   it   t ril iix>"rni'iti»n   "t'  tlif  Kt-f. 

I::   !_T(-!iera!   tlie  delinitioii   of  a  '_;-i'oiip  of  dperat  ions  involves  also 

i  onditii'iis  that   tin'  operations  shall  In1  as.soeiat  ivr  ;ind  that  the 

ideiitieal    t  ran-fonnat  ion    shall    lie  delined.     These   latter  eoiidit  ions 

-    true    for    <j;(.,,inetriea!    transformations   need    not    lie 

:   in  our  iletinit  ion  nor  explicit  Iv   looked   I'm1  in  (k'tt'i'inining 

\\hethei-  MI   not  a  -_;'!\'e!i  set  nf  transformations  form  a  -^roiip. 

\s  an  example  ol    a  "I'mip  consider  the  operations  consist  in""  of 

i  .— i 

its   in  >paee  around  a  tixed  axis  throii^li  anv  ang'le 

.    -1-TT 


ar<    ind  t  lie  same  a\  is. 

Ionium.;'  a   '_:i«np  and   contained   in   a  larger 
/'  "t    the  larger  'jroiip.     l-'nr  example,  the  rota- 


(IKNKRAL  CONCEPTS  7 

group  of  mechanical  motions.  All  translations  in  a  fixed  direction 
form  a  subgroup  of  the  group  of  translations  and  hence  a  sub- 
subgroup  of  the  group  of  motions. 

The  importance  of  the  concept  of  groups  in  geometry  lies  in  tin- 
fact  that  it  furnishes  a  means  of  classifying  different  systems  of 
geometry.  The  element  of  the  geometry  haying  been  chosen,  any 
group  of  transformations  may  be  taken,  and  the  properties  of 
geometric  figures  may  be  studied  which  are  unaltered  by  all  trans- 
formations of  the  group.  Thus  the  ordinary  geometry  of  space 
considers  the  properties  of  figures  which  are  unaltered  by  the  group 
of  mechanical  movements. 

Any  property  or  configuration  which  is  unaltered  by  the  opera- 
tions of  a  group  is  called  an  incuriniit  of  the  group.  Thus  distance 
is  an  invariant  of  the  group  of  mechanical  motions,  and  a  circle  is 
an  invariant  with  respect  to  the  group  of  rotations  in  the  plane 
of  the  circle  about  the  center  of  the  circle. 

EXERCISES 

1.  It'  ./•  is  the  distance  of  a  point  P  on  a  straight  line  from  a  fixed 
point  a,  and  /'  is  transformed  into  a  new  point  />(  such  that  x'  =  ».r  -f-  l>, 
prove  that  the  set  of  transformations  formed  bv  yiving  to  n  and  //  all 
possible  values  form  a  i^roup. 

2.  I  f  i ./•,  // 1  are  <  'artesian  coordinates  in  a  plane,  and  a  trans  format  ion 
is  expressed  l>v  the  equations 

.<•'=  j~  cos  a  —  i/  sin  a, 
'/'=  .''  sin  ic  -|-  //  cos  <t, 

prove  that  the  transformations  obtained  by  t;-iviiiL,r  n  all  possible  values 
form  a  ^roup. 

3.  It'   (./•,    i/}   are    ('artesian    coordinates    in    a    plane,    prove    that    the 
transformations   defined   by   the  equations 

./•'  =  j-  cos  K  -f  ij  sin  n, 

//' '=  ./•  sin  T  —  //  cos  (t, 
do   not    form   a    <_rroii]>. 

4.  Name  Mime  subgroups  of  the  Croups  in  Kxs.  1    2. 

5.  Let   f,'  be  a  i;iven  L;TOH|>  and  <!    a  subgroup.     It'  every  transforina- 
tioii  of  'r't  is  re|)laeed  bv  its  transform  bv  7'.  where  T  belongs  to  '/.  show 


CHAPTER  II 

RANGES  AND  PENCILS 

7.  Cartesian  coordinate  of  a  point  on  a  line.    Consider  all  points 
\\hich  lie  on  a  line  A  A' (Fig.  1).    These  points  are  called  a  jH-ncil 
or  a   r<i/i//>',  and   the   hue    LK  is  called   the  </.r/,s  or  the   Ixittf  of   the 
range.    Any  point    /'  on   LK  may  4  o        f 

lie  fixed  most  simply  by  means  of  ^ 

KM,.  1 
its  distance  of  from  a  fixed  origin 

o,  the  distance  being  reckoned  positive  or  negative  according  as  /' 
lies  on  one  side  or  another  of  O.  We  may  accordingly  place 

x=OP  (1) 

and  call  .r  the  coordinate*  of  -/-".  To  any  point  /'  corresponds  one 
and  only  one  real  coordinate  .r,  and  to  any  real  j-  corresponds 
one  and  only  one  real  point  P.  Complex  values  of  ./•  are  said,  as 
in  sj  8,  to  define  imaginary  points  on  LK. 

The    coordinate    may    be    made    homogeneous    ($  4)    by   using 

the  ratio  ./• :  f.  where  -  =  Of.    As  /'  recedes   indefinitely  from   O,  t 

f 

approaches  the  value  0.  Hence,  as  in  £  4,  we  make  the  convention 
that  the  line  has  one  point  at  infinity  with  the  coordinate  1  :  0. 
When  the  nonhomogeiieons  j-  of  (1)  is  used,  the  point  at  infinity 
lias  the  coordinate  s,. 

Tlu-  coordinate  ./•  we  call  the  Cartesian  coordinate  of  /'because 
ot  its  familiar  use  in  ('artesian  geometry. 

8.  Projective   coordinate   of  a.  point  on  a   line.    On   the  straight 
line    LK  (Fig.   1)   assume   two   fixed   points  of   reference   A   and    /»' 
and   two  constants  /-    and  /ro.     Then    if   /'  is  any   point   on   LK  \ve 
may  take  as  the  coordinate  of   /'  the  ratio  ./•  :  ./•„,  where 

V^^,  •-•!/':  V  /;/'<  0) 

•  The  wop!  "  riii'iriliiiatc  "  may  lie  objected  tn  nil  the  ground  that  it  implies  the 
e\ist'-iice  ,,|'  l([   1,-ot  twn  quantities  which  an.-  cnnnlinatecl  in  the-  usual  sense.     In 

Spite  nt  thi>  nlijei'tinn  we  retail!  the  Word  tn  elnpha>i/.e  the  fact  that  We  liave  hel'e 
ti»-  >iin]ile>!  case  of  coiii'dinates  in  an  u-diliiensioiial  tceoinetry. 

s 


KANtJKS  AM)   PKNCILS  y 

hi  which  the  distances  AT  and  7>'/'  are  positive  or  negative  accord- 
ing as  /'  is  on  the  one  side  or  the  other  of  A  or  />'  respectively. 
It  is  evident  that  the  correspondence  between  real  points  on  IJ\ 
and  real  values  of  the  ratio  j^  :.rt  is  one  to  one.  Complex  values 
of  the  ratio  define  imaginary  points  on  LK  ($^). 

The  Cartesian  coordinate  of  the  preceding  article  may  he  con- 
sidered as  a  special  or  limiting  case  of  the  kind  just  given.  For  it 
in  (  1  )  \ve  place  /-  —  1,  allow  the  point  1>  to  recede  to  inlinitv, 
and  at  the  same  time  allow  /".,  to  approach  zero  in  such  a  manner 
that  the  limit  of  //.,  •  /•'/'  remains  finite,  equations  (1  )  give  the 
homogeneous  Cartesian  coordinates  of  ]'. 

Considering  (1),  we  see  that  as  /'  recedes  indefinitely  from  .1 
and  /.'  the  ratio  ./',:./'.,  approaches  the  limiting  ratio  k^ :  k^.  Hence 
we  sav  that  the  line  has  one  point  at  infinity. 

It  is  to  l»e  noticed  that  the  ratio  (which  alone  is  essential)  of 
the  constants  k^  and  /-o  is  determined  by  the  coordinate  of  anv  one 
point.  Since  this  ratio  is  arbitrary  the  coordinate  of  anv  point  mav 
be  assumed  arbitrarily  after  the  points  of  reference  are  fixed. 

In  particular  anv  point  may  be  given  the  coordinate  1  :  1.  This 
point  we  shall  call  the  unit  j><>int.  The  coordinate  of  A  is  <>  :  1  and 
that  of  /.'  is  1  :  0.  Since  the  unit  point  and  the  points  of  reference 
arc  arbitrarv,  ii  follows  that  tit  scttint/  uj>  tltc  coordutntc  *//*/<//;  ti/tj/ 
flirt  f  jio/'/itx  ni'ii/  In'  i/ti't'/i  (I/,-  coord  iiKiti'H  (>  :  7,  J  :  O*  <tn<l  /  :  /  /v.vy^-c- 
tici-l//,  it/id  t/n'  i-oo/'diiKftt"  xyxtt'in  in  fully  di'tcnnini'd  /<//  tht'Kt1  i>oint#. 

The    coordinate    of    this    section    we    shall    call    the    projective 

mlinate   of   /'  because   of  its   use   in   projective  geometrv. 


CO 


EXERCISES 

1.  I-'>t;ilili>li  ;i  coordinate  svMrin  on  a  straight  line  so  that  tin-  (mint  /.' 
i>  ~>  iiiclit-s  to  tin-  1'i^'lit  of  .1  and  tin-  unit  point  1  inch  t  o  1  he  ri^ht  of  .  I . 
\Vhciv  is  tin-  coordinate  negative'/ 

'2.  Take  the  point.  /;  as  in  MX.  1  and  the  unit  point  1  inch  to  the 
ri^'lit  of  /;.  \Yhat.  are  the  coordinates  of  points  respect  ivel\  1,  L'.  .'!, 
•1  inches  to  the  ri^ht  of  .!  and  1,  L',  M  inches  to  the  left  of  .1  '.' 

9.  Change  of  coordinates.  The  most  general  change  from  one  svs- 
tcin  ot  projective  coordinates  to  another  mav  he  made  hv  changing 
the  points  of  reference  and  tin-  unit  point,  the  latter  change  beiiiii 

I  I*) 

ei jui\  alent    to   clianu'm^   the    ratio   of   the    constants   /,•     and    /,,,.      Let 


10  ONE-DIMENSIONAL  (1KOMKTUY 

./•  :  ./-.,  IK-  the  coordinate  of  any  point  /'  (Fig.  -)  referred  to  the 
points  of  reference  .1  and  />.  with  certain  constants  k^  and  /r,,  and 
let  J\  '  .'•',  he  the  coordinate  ot  the  same 
point  referred  to  the  points  ol  re  I  ere  nee  ^ 
A'  and  /•'',  with  constants  k\  and  k'.,. 
Assume  any  point  <>  and  let  (>A  —  <i, 

o.l'     <i'.  (Hi  —  f>.  oil'     I',  and  ^/'      (.    Then  from  (1"),  §  8,  we  have 
j-l:j:,^kl{t-«):k.,(t    -/'),      -'I  :  -<  -  --k((t  -  a'):  k',(t  -//).     (1  ) 
The  elimination   of  /  Iroin  these  equations  <_dves  relations  of  the 
form  pj.}    raX  +  rtX' 


which  arc  the  retjnired  toi'mnlas  lor  the  change  of  coordinates. 

'1'he  i-atio  of  the  coet'ticiciits  n^  a;,,  (3^  and  /:?,  will  lie  determined 
if  we  know  three  values  of  rt  :  ./•.,  which  correspond  to  three  values 
of  .r[  :./•'.,  in  particular  to  the  three  values  0:1,  1:0,  1  :  1.  For 
when  ./•[  :  ./•',  —  "  :  1  \\  e  have  j\  :  :r.,  =  a.,  :  /^.,;  when  ./•,'  :  ./•.'  =  1:0  we  have 
./•,  :  ./•„  --  n  ,  :  /:?,  :  and  when  ./•(  :  .r',  --1  :  1  we  have  .r,  :  .r.  ,  =  rr,  +  a'.,  :  /rf,  +  /^.,. 

It  is  obvious  from  the  foregoing  that  if  the  reference  points  A 
and  '//are  distinct,  the  coefficients  in  (I?)  must  satisfvthe  condition 
n  /^--a.,^?  --  0,  which  is  also  necessarv  in  order  that  the  ratio  ./'  :./•, 
in  e(|iiations  ('!)  should  contain  ./•,:./•.',. 

lOipiations  (  '1  )  niav  lie  placed  in  a  form  whieli  is  of  frequent  use. 
Let  ns  place  ,'-J  :  .r',  \,  n^  =  2^  /^]  =  ,r,,  n  ,=  //^  /3.,=  //.,.  \\'here  //(  :  //., 
and  ,:-•  :  ,?,  are  the  coordinates  ot  the  two  points  corresponding  to 
A.  :  0  and  X—  s.  respectively.  Then  equations  (-)  liccome 


Ilciicc,  /'/'  //   :  //o  <nnl  ,r   :  ,r,  nrr  f/n'  <-nf!i'iTi  unit's  of  any  f/m  piiintu  <>n 
(i    xtrn'ujlit     Inn',    f/n-    fix'SrtJinrttt'    <>f  <<ni/   <>tlnT    /»>///f    tun//   In-    U'rittrn 

;i   •  *.-',;  //.,  f  X.?.,. 

EXERCISES 

1  .    I-'ind  t  he  formulas  for  t  he  change  from  the  coordinate  in  Kx.  1  ,  >  S, 
to  that   in   I''.\.  L'. 

'.:!.    l''md  the  |ormula-<  fur  a  change  from  the  coi'u'dinate  in   I'lx.  1,  *  S. 

''>    Olie    in     \vllich    tin-     ri'l'i'Trlii-e     jiollits    a  I'e    1'eS]  icet  1  \"e!  \'    L'    ami     t)    inches 

I'l'oin   .1   ami  the  unit   point    1   units  from  .  I  . 

.''  .    l'v<''.e  that  all  changes  ot  coord  inatfs  foi-m  a  L;TOU)P. 


IlAN<;i-:S   AND    I'KNCILS 


11 


10.  Coordinate  of  a  line  of  a  pencil.  Consider  all  straight  lines 
which  lie  in  a  plane  and  pass  through  the  same  point  (Fig.  '•'>). 
Such  lines  form  a  jH-m-i/,  the  common  point  being  called  the  ?•>'/•/••./• 
of  the  pencil. 

Let  <>M  be  a  fixed  line  in  the  pencil.  <>/'  any  line,  and  0  the  angle 
Mnl'.  Then  it  would  be  possible  to  take  0  as  t he  coordinate  of  <>l\ 
but  in  that  case  the  line  <)/'  would  , 

have  an  infinite  number  of  coordi- 
nates differing  by  multiples  of  '1  IT. 
We  may  make  the  relation  between 
a  line  and  its  coordinate  one  to  one 
by  taking  as  the  coordinate  a  quan- 
tity JC  detined  by  the  equation 

//")  /-  -|     N 

•tan0,  (1) 

where    k   is    an    arbitrary   constant. 

Then  ./--I)  is  the  line  ".)/,   ,/•  =  -/-_  is 

the  line  at  right  angles  to  O.17,  and 

any  positive  or  negative  real  value  of  ./•  corresponds   to   one   and 

only  one  real  line  of  the  pencil,  and  conversely.     Imaginary  values 

of  ./•  define  imaginary  lines  of  the  pencil  as  in  Jj  3. 

A  more  general  coordinate  may  lie  obtained  by  usin^  two  lixed 
lines  of  reference  <>A  and  nil  and  defining  the  ratio  ./•  :./•.,  by  the 
equation  .,-  .  ,-it= /{•  sin  JO/J :  £.,  sin //o/'.  c2) 

Equation  ('2)  reduces  to  equation  (  1  )  when  the  an^le  Anil  is  a 
ri'_dit  angle,  n_\  coincides  with  <>M,  and  ./•  :./•.,  -  r. 

In  general  let  the  angle  M<>.  \  ....  n  and  the  angle  .]ft  >/; -=  rf.  Then 
(.! )  may  be  writ  ten 

s:j\t=/r  sin  (  0  --  n  )  :  /'.,  sin  (  0  -  $ ) 

=  /-(./•  cos /{     - /•  sin  '():/'(.'•  cos  /^    -  /,'  sin  /3\  (  :>>  ) 

1  '2 

\\'hen  .r  is  defined   liy   (  1  ). 

Now  let  ./-|  :  ./•.',  be  another  coi'irdinate  of  the  lines  of  the  pencil  of 
the  same  form  as  in  equation  ('2).  but  referred  to  lines  of  reference 
< >A'  and  nil'  and  with  constants  l,-'{  and  /•'.  Then  ./ ,'  :  ./^  is  connected 
with  ./•  :  ./•  by  a  bilinear  relation  of  the  form 


1'J  (>Ni:    D1MKNSIONAI,   CKO.MKTKY 

This  follows  from  the  fact  that  both  .r,  :  .r,  and  ,r[  :  j\,  are  con- 
nected \\ith  ./•  bv  a  relation  of  the  torm  ('•*>). 

Since  a  transformation  of  coordinates  is  effected  either  by  change 
of  the  lines  of  reference  or  bv  change  ot  the  constants  /"  ami  /-o,  it 
follows  that  anv  transformation  ot  coordinates  is  expressed  by  a 
relation  of  form  ( 4~).  The  coefficients  of  the  transformation  are 
determined  when  the  values  ot  J\:  J\,  are  known  which  correspond 
to  three  \alues  of  ./ [  :  r.,.  The  proof  is  as  in  sj  1'.  Also,  as  in  >j  '.*,  it 
mav  be  shown  that  if  //,://„  mid  ^  :  .?,  are  the  coordinates  of  anv 
two  lines  of  a  pencil,  the  coordinate  of  any  line  may  be  written 


11.  Coordinate  of  a  plane  of  a  pencil.  Consider  all  planes  which 
pass  through  the  same  straight  line  (Fig.  4).  Such  planes  form  a 
}i,i,'-il  or  xln-iif,  and  the  straight  line  is  called  the  n.n'x  of  the  pencil. 
The  coordinate  of  a  plane  of  the  sheaf  mav  be 
obtained  bv  first  assuming  two  planes  of  refer- 
ence it  and  /•  and  a  fixed  constant  k.  Then,  if  j> 
is  any  plane  of  the  pencil  and  (>/,  j>~)  means  the 
angle  between  </  and  />,  we  mav  define  the  coordi- 
nate of  y>  as  the  rat  io  ./•:  .r  given  bv  tht1  equations 

•'",  :  •''.,    :  k  sni  ( ''-  )> )  '•  /•".,  sin  ( ^.  }>  )•          ( 1  ) 

It  is  obvious  that  if  a  plane  ///  be  passed  per- 
pendicular to  the  axis  of  the  pencil,  the  planes  of 
the  pencil  cut  out  a  pencil  of  lines  in  the  plane  m. 
The  angle  between  two  lines  of  this  pencil  is  the 
plane  angle  ot  the  two  planes  in  which  the  two  lines  he.  Hence 
the  coordinate  .'',:.''.,  defined  in  (1)  is  also  the  coordinate  of  the 
lines  of  the  pencil  in  the  plane  />/,  in  the  sense  of  ^  10.  The  results 
ot  ;j  10  with  retereiice  to  transformation  of  coordinates  hold,  there- 
fore, for  a  pencil  of  planes.  In  particular,  if  //,://.,  and  z^.z,,  are 
the  coordinates  ot  anv  two  planes  of  a  sheaf,  the'  coordinate  of  anv 
plane  '  'f  t  he  pencil  mav  be  writ  ten 


Fit;.  1 


p.'-       //,  +  X.?.,. 


CHAPTER  III 

PROJECTIVITY 

12.  The  linear  transformation.  We  shall  now  consider  the 
substitution 

('*!#,  —  <r  ,/tfj  —  0  )  ( 1  ) 

pj'..  =  't,./- ,  -fpV, 

not  as  a  change  of  coordinates,  as  in  ij '.',  but  as  defining  a  trans- 
formation in  the  sense  of  ^  •>.  Then  .i\:.r,  are  to  be  interpreted  as 
the  coordinate  of  an  element  of  a  one-dimensional  extent  and 
.>•,':./•.'  as  the  coordinate  of  the  transformed  element  of  the  same  or 
another  one-dimensional  extent.  If  J\1J'.,  and  .r|  :  ./•'  refer  to  dif- 
ferent extents,  the  elements  need  not  be  of  the  same  kind.  For 
example,  the  transformation  (1)  may  express  the  transformation 
of  points  into  lines,  of  points  into  planes,  of  lines  into  planes,  and 
so  on. 

To  study  the  transformation  we,  shall  lind  it  convenient  to  use 
a  nonhomogeneous  form  obtained  bv  replacing  .r,  :  .r,  by  X.  .i'[:j'^ 
by  X',  and  (.'hanging  the  form  of  the  constants.  We  have 

\'  -  .  (n&  —  rfy  '--  0  )  ( -2) 

Here  X  and  X'  may  be  the  point,  line,  or  plane  coordinates  of 
^  7,  H,  10,  11  or  may  be  the  X  used  in  the  formulas  of  ^  t>  1  1. 
More  generally  still,  X  may  be  any  quantity  which  can  he  used 
to  define  an  element  of  any  kind,  even  though  not  vet  employed 
in  this  text. 

In  each  case  the  clement  with  coordinate  X  is  said  to  be  trans- 
formed into  the  element  with  coordinate  X'.  and  the  two  elements 
X  and  X'  are  said  to  correspond.  There  is  one  and  only  one  element 
X' corresponding  to  an  element  X.  Conversely,  from  (-)  \\  c  obtain 

X  =      'X    ,  •  '     •  (  :'. ) 

i:; 


14  <>XK  imiKxsioxAL  <;KOMKTRY 

Hence  to  an  element  X' corresponds  one  and  only  one  element  X. 
In  other  words,  the  correspondence  between  the  elements  X  and  the 

Any  clement  whose  coordinate  is  unchanged  by  the  trans- 
formation is  called  a  fired  element  of  the  transformation.  This 
definition  has  its  chief  significance  when  the  elements  X  and  X'  are 
points  of  the  same  range,  or  lines  of  the  same  pencil,  or  planes 
of  the  same  pencil.  If,  for  example,  X  and  X'  are  points  of  the 
same  range,  the  point  X  is  transformed  into  the  point  X',  which 
is  in  general  a  different  point  from  X,  but  the  fixed  points  are 
unchanged. 

To  find  the  fixed  elements  we  have  to  put  X  —  X'  in  (2)  or  in  (3). 

There  results 

7X'J  +  (8  -  a  )  X  -  /3  =  0.  (4) 

Any  linear  transformation  has,  accordingly,  twofijred  elements,  which 
may  f>e  distinct  ,\r  coincident. 

If  '«,  /3,  7,  and  6  are  real  numbers,  and  real  coordinates  X  and  X' 
correspond  to  real  elements,  we  may  make  the  following  classifica- 
tion of  the  linear  transformations: 

( 1  )  (  8  —  a)2-f-  4  fiy  >  0.  The  fixed  elements  are  real  and  distinct. 
The  transformation  is  called  hyperbolic. 

('2)  (8  —  a  )2-f  4  $7  <  0.  The  iixed  elements  are  imaginary  with 
conjugate  imaginary  coordinates.  The  transformation  is  called 
elliptic. 

( -\ )  (  6  —  a  )•+  4  #7  =  0.  The  fixed  points  are  real  and  coincident. 
The  transformation  is  called  parabolic. 

P>\  the  transformation  (2)  an  element  /'  with  coordinate  X  is 
transformed  into  an  element  <t>  with  the  coordinate  X'.  At  the 
same  time  the  element  ((>  is  transformed  into  an  element  l>  with 
coordinate  X".  In  general,  I!  is  distinct  from  /',  for  X"  is  given 
by  the  equation 

X"  —  --•  ( •"> ) 

In  order  that  X"  should  always  be  the  same  as  X  it  is  necessary 
and  sufficient  that  the  e<|iiat ion 


PRUJKCTIVITY  Jo 

should  be  true  for  all  values  of  X.     The  coeilicients  a,  $,  7,  and  6 
must  then  satisfy  the  equations 

iiy  +  78  =  0, 


The  second  equation  gives  6  =  ±  a.  If  \ve  take  8  —  a  the  other 
two  equations  give  7  =  <*,  /^=  0,  and  the  transformution  (  1  )  reduces 
to  the  identical  transformation  \=\'.  We  must  therefore  take 
8  =  —  rr,  and  all  three  equations  (t!)  are  satisfied. 

The  transformation  then  hecomes 


0)  (7) 

7  A.  —  ft 

A  linear  transformation  of  this  type  is  called  hn-ohifori/.  It  has 
the  property  that  if  repeated  once  it  produces  the  identical  trans- 
formation. The  correspondence  between  the  elements  X  and  the 
transformed  elements  X'  is  called  an  invAutiun. 

EXERCISES 

1.  Find  the  transformation  which  transforms  0.  1.  x  into  1,  x.  0, 
respectively.    AYhat.  are  the  fixed  points  of  the  transformation  '.' 

2.  If  ./•   is  the   Cartesian   coordinate  of  a   point  on   a   straight   line, 
determine  the  linear  transformation  which  interchanges  the  origin  and 
the  point  at  infinity.    What  are  the  fixed  points  of  the  transformation  '.' 
I  >o  all  such  transformations  form  a  group'.' 

3.  If  jr   is   the  Cartesian   coordinate   of  a   point    on   a   straight   line, 
determine   the   transformation    which    has   only    the   origin    for  a    fixed 
point  and  also   that    which    has   onlv  the   point   at    inlinitv    for  a    fixed 
point.     Does  each  of  these  tvpes  of  transformation  form  a  group'.' 

4.  1  f  x   is   the   Cartesian   coordinate   of   a   point    on   a   straight    line, 
determine  a  transformation  with  the   fixed   points  4-  I.     Po  these   form 
a  group  ? 

5.  Show  that  the  general  linear  transformation  mav  he  oht 
the    product   of   two   transformations   of   the   type   X'  =  "A.    tw 

type  X'  =  X  -f-  />,  and  one  of  the  tvpe  X'  =  -• 

A 

6.  Show  that   anv  transformation   with  two  distinct    fixed  elements 

X'  —  "  X    -  '/ 

"  and  It  can  he  written  -       -    =  I; 


X   —b  X  -  h 


1C,  ONK   DIMENSIONAL  CKO.M  KTKV 

7.  Slm\v    that    any    transformation    \vitli    a    single    fixed   element    a 

can   he   writ  ten  4-  /'• 

A  —  "       A       " 

8.  Show     that     anv     involutorv     transformation     can     he     written 

•   where  "  and  l>  are  the  fixed  elements. 
A'       /•  A    -  I' 

it.    Shiiw    that    all    transformations    with   the    same    fixed   elements 
form   a    urroup. 

10.  < 'on>ider   the   set    of  circles    which    pass   through   the   same   two 
tixed   poinN.  and  the  common  diameter  of  the  circles.     ShoW  that   if  /' 
and    i.>  an-  the  two  points   in    which  any  one  of  the  circles  meets  the 
common    diameter,    /'    mav    lie    transformed    into    <>    bv   an    involutory 
transformation,  the  form  of  which  is  the  same  for  all  points  /'.    Show 
that  the  transformation   is  elliptic,  or  hyperbolic  according  as  the  two 
fixed  points  in  which  the  circles  intersect  are  real  or  imaginary. 

11.  Show,  conversely  to  Kx.  10,  that  any  involutory  transformation 
mav  he  geometrically  constructed  as  there  descrihed. 

13.  The  cross  ratio.  Tlie  linear  transformation  contains  three 
constants:  namely,  tin'  ratios  of  the  four  coefficients  a.  tf,  7.  and  8. 
These  constants  can  he  so  determined  that  anv  three  arbitrarily 
assumed  values  of  A  can  he  made  to  correspond  to  anv  three  arbi- 
trarilv  assumed  values  of  A'.  In  other  words, 

7.  /•'//  it  liiifiir  trcinxfurination  anif  tJirff  rloncnfn  <•<'//  f>i-  trnnitfnnncff 
'nit"  iini/  nf/i,'/'  tfn'i'i'  t'li'/iu'ittx,  ft n<f  tJifxi'  t/irt'f  !>///)•!<  i >f  I'orrfKpnnilinfj 
>  li-nii'iit*  <it'f  soitfif'it'nt  t"  ft.r  flic  trnn»f»rinnt\<in. 

'I'n  \\rite  the  transformation  in  terms  of  the  coordinates  of  three 
pairs  of  corresponding  elements,  we  write  first 

A'      \,  A  -  A, 

,       '(  -•  (  1  ) 

A  -  A;       A    A, 

\\'hicli  i<  obvioii>l\-  a  transformation  bv  which  A(  is  transformed 
into  A,',  and  A,,  into  A',.  If.  in  addition.  A,  is  to  be  transformed  into 
A  .  '(  must  be  determined  bv  the  eipiatlon 

A.' .  -  A'          A,      A, 

;    =  ^     '  — S-  cl) 

A  .   -    A .  A.,  —  A. 

Kr«  mi    (1  )   and    (  '1  )  we   ha  ve 

\'    A:   A;    A;    A  -  \n_  A,-  A, 

A'       \[  '  \'       A;       \    -  A,  '  A.       A., 
which  i>  tlic  rciiircil  transtormatioii. 


I'KO.IKCTIVITY  17 
If  X  iuid  Xj  are  a  fourth  pair  of  corresponding  elements,  we  have, 

from  (o),         x;  -  x;   x:;-x;    x4-x,  x,-x, 

A,  —  A..      A.,  —  A,,        A,    '~  A,  A ,  •—  A., 

or,  with  a  slight   rearrangement, 

X.  —  A,      A,     -  A,        X,  —  X,  X, —  X. 

••        ~_         § i         •>  _          ^                              f  i  \ 

X    —  X       X ,        X.         X    —  X  X   —  X 


X,-  X(     X.,  -  X, 

is  called  the  r/v/.s-x  ratio,  or  the  iiiilniDiinitfc  ratio,  of  the  four  ele- 
ments X(,  X,.  X.(,  X(,  and  is  denoted  by  the  symbol  (\\t.  X^). 
Kquation  (4)  establishes  the  theorem: 

II.  Tin1  t't'oxx  ratio  (if  fniir  clt'tticnta  ix  u/ia/fi'm/  hi/  a////  lun-ar 
t  ra  iixiorniat  inn. 

The  cross  ratio  is  accordingly  independent  of  the  coordinate 
system  used  in  defining  the  elements. 

The  cross  ratio  depends  not  only  on  the  four  elements  involved 
but  also  on  the  order  in  which  they  are  taken.  Now  four  things 
maybe  taken  in  twenty-four  different  orders,  but  there  result  only 
six  distinct  cross  ratios.  In  fact,  it  is  easy  to  show,  by  writing  all 
possible  cross  ratios,  that  the  six  distinct  ones  are 

1  1  /-I 

/•,  1  —  /•,  •  •  i 

/  1  III  1 

where  /•  is  any  one  of  t  hem. 

In  naming  the  cross  ratio  of  four  elements  it  is  therefore  neces- 
sary to  indicate  the  order  in  which  the  elements  are  to  be  taken. 
\Ve  have  adopted  the  convention  that  if  /',  /!.  /',  and  /[  are  four 
elements  with  the  coordinates  X^  X,,  X..  and  X(  respectively,  the 
cross  ratio  indicated  by  the  symbol  (/'/',.  /'/,')  shall  be  <_nven  by 
the  relation 


18  <>XK   DIMKXSIOXAL  CEOMETRY 

A  special  form  which  the  cross  ratio  takes  for  certain  coordinates 
is  of  importance  and  is  given  in  the  following  theorem: 

///.  If  tin-  i-lfim-ntx  /'  a  ml  <t>  h<n->'  tin'  foordinatt'x  i/{ :  //„  <tn<1  ^ :  zn  >v- 
nih'i-ffr,  ///,  ,1ml  tin-  >l>  //n  -ntft  A'-///'/  X  /t/iri-  the  coordinates  //  4-  X^  :  //.,  4-  \z 
itihl  >/  4-  fj..:  :  //., 4-  H- ,  /vx/'fv//>r/ty,  tln'n 

(  /'<,>.  /,'.s'  )-    (  A' A',  /'(,>)  =  -' 

To  prove  this  take  X  ="  for  the  element  /',  X,=  o;  for  the 
element  (t>,  \  -X  for  the  element  //,  and  X^=  ^  for  the  element 
N,  and  substitute  in  ( <i ). 

If  X  is  the  Cartesian  coordinate  of  a  point  on  a  straight  line, 
then  Xj  —  X3= /'/'.  X,— X4=  /4'/,',  X_,—  X.,  =  !'.!',,  X.,—  X4=  /4A_f,  and 

/I>/-''     7^/4> 

The  cross  ratio  is  accordingly  found  l>v  finding  the  ratio  of  the 
segments  into  which  the  line  I'./j  is  divideil  li\-  /,'  and  the  ratio  of 
t  he  segments  into  which  A'/,'  is  divided  l>v  /,',  and  forming  the  ratio 
of  these  ratios. 

14.  Harmonic  sets.  If  a  cross  ratio  is  equal  to  —  1,  it  is  called 
a  harmonic  /•<///</.  It  /',  /',,  /',  and  l\  are  four  elements  such  that 

(/;/.;,  /•/;>=  ~1, 

the  four  elements  form  a  harmonic  set,  and  the  points  7J  and  P, 
are  said  to  he  harmonic  conjugates  to  A|  and  /4'. 

From  III,  ^  1  o,  it  follows  that  the  points  //)  4-  X,^  :  //.,  4-  X^,  and 
//,  -  X^://,  -  X,r,  are  harmonic  conjugates  to  //,://.,  mid  z  :  zn. 

I-'rom  (7),^  1  •>,  it  follows  that  if  four  points  on  a  straight 
line  toriu  a  harmonic  set,  then 


'I  his  shows  that  the  two  points  in  a  harmonic  set  divide  the  dis- 
tance between  their  harmonic  conjugates  internally  and  externallv 
in  the  same  rat  n >. 


1'KO.JKCTIYITY  111 

EXERCISES 

1.  Show  that   tlir   cross   ratio  of  any   point,  the   transformed    point, 
and  the  two  tixed  points  ot    anv  elliptic  or  hyperbolic  transformation 
is  constant.    This   is  sometimes  called   the  rhui'artm-ixt'ii'  OV/NS  ,-ntln  of 
the  transformation.     What    happens  to  the  characteristic  cross  ratio  as 
the  two  tixed  points  approach  coincidence  '.' 

2.  Show    that,    \\\    any    involutory    transformation    anv    element     is 
transformed  into  its   harmonic  conjugate  wiih   respect  to  the  two  fixed 
elements. 

3.  If  AI?  A.,,  A(,  A(  form  a  harmonic  set.  prove  that 

1'  1  1 


In  general,  prove  that   if  lAjA.,,  A.(Aj  =  //, 

1  -  /.-  1  _/,-_ 

\-\\-X    \-\ 

4.  \\'rite  the  transformation  l>y  which  each  ]ioint  on  a  line  is  trans- 
formed into  its  harmonic  conjugate  with  respect  to  the  points  A=   —  ", 
A  =  ".     What  are  the  tixed  points  of  the  transformation  ". 

5.  Prove   that    an    involution   of   lines   of  a    pencil   contains   one  and 
onlv  one  pair  id'  perpendicular  lines  (that    is,  one  case  in  which  a  line 
is  perpendicular  to  its   transformed   line)  unless  all   pairs  of  lines  are 
perpendicular.     When  does  the  latter  case  occur'.' 

(3.    Let  ,''t  :  .'•„  lie  the  coordinate  of  a  point  on  a  line  and  consider  the 
point  pair  defined  by  the  equation 

wj'"  +  -  ".'''   +  "•••••''•   =  0. 


7.  Let  A  and  /.'  be  two  distinct  points  defined  bv  the  equation  of 
Kx.  l>,  ami  /'  (i/  :  //., )  and  (j  (  :  :  :: , )  and  I!  (tr  ;  n\, )  any  t  hree  points.  If 
the  jn'ujt'ctlrt'  tlittfiun'f  between  two  points  is  defined  bv  the  equation 


( 'onsider  t  \\'i i  cases  : 

1.  .1  and  /.'  real.  Take  /.-  real.  Then  any  two  points  between  .1  and 
l>  ha\'e  a  real  distance  apart.  I  and  /.'  are  at  an  infinite  distance  liom 
an\  other  point.  An\  point  not  between  I  and  /.'  is  at  an  imaginary 
distance  from  an\  point  l>et  \\een  I  ami  /.'. 


'20  OM-:-I)IMKNSIONAL  (JKOMETUY 

L'.  .1  ami  /.'  conjugate  imaginary.  Take  I;  pure  imaginary.  Any  two 
ival  jtuiiits  arc  at  a  ival  liiutt-  distance  apart.  The  total  length  of  the 
line  i>  tinite. 

8.    ('(insider  the  point   pair  detined  by  the  equation 

"n-''i"  +  -  "i--'Y''j  -f  "•-.'••;  =  0. 
1'heii,  if  //j  :.'/.,  is  any  given  point,  the  equation 


defines  a  point  \vldeh  is  called  the  j/n/nr  /mi/tt  of  //  with  respect  to 
the  point  pair.  Assuming  <tr/'.,._,  —  cfj  :-'-  0,  show  that  to  any  ]>oint  cor- 
responds a  definite  polar  point  and  that  any  point  is  the  polar  point 
of  a  definite  point  //.  Show  that  a  point  and  its  polar  are  harmonic 
conjugates  with  respect  to  the  point  pair.  What  happens  to  these 
theorems  if  "n".^..  —  "f.  —  0  '.' 

15.  Projection.  Two  Diu'-dhnensioiial  extents  are  said  to  lie  in 
proji'i-tinn  if  the  elements  of  the  two  extents  are  brought  into 
correspondence  by  means  of  a  linear  relation, 


between  their  ci  litrdiiiates.  The  correspi  unleiice  is  called  a  i>i''>/cc- 
tii'iti/.  It  the  correspundence  is  inyolutory,  the  proji'ctivitv  is  an 
in\'olutit)ii  f^ll').  I-'roin  the  definition  the  following  theorems 
may  be  immediately  deduced  : 


II.  /'//'"   'iiii'-il  I  iiti'iixtoii/il  r.rfriifn   itnlif  In'   /'/•"lli/J/f   iiit'i  l>r"i>'i'f  l"H    K'ltJl 
i  '/<•/!    otJtt'f    til    aiii'li    it    //'it//  tlnlt    n/i/l    ttn'i-i'    >-lt'iiti'ittx    nt'  n/ii'    i<r>'    //nlifi'    fu 

<'nri't'xp<iiid  t"  iinif  tln'i-i'  iti'nii  tit*  i,t'  //t,1  nf  /n'/\ 

III.  A  (if'i'ii'ffirfti/  /.s  full  if  il<  ti  ruti  lli'il  In/  f/i/'i'/'  y"//Vx  "/'  I'li/'ft'ttiiittlifi'tli/ 

,  /,  ,//,,/  f,: 

IV.  Tit'n   i.rtiiit*   //•///,-//   I//-,-    in    ii/-,i/i'i'/  tu/t    tt'lt/i    flu'   ft  tin'   tin  /'d   r.rtiiit 

il/'f     III     [ll'i'l  li'tlo/l      l/'tf/l      1'ili'JI      n/l/i/-. 

EXERCISE 


PKOJKCTIVITY 


21 


\ 


16.  Perspective  figures.  A  simple  case  of  a  projeetiyity  is  that 
called  a  perspectivity,  now  to  he  ill-lined.  Noting  that  we  have  to 
do  with  pencils  of  different  kinds, 
according  as  they  are  made  up 
of  points,  lines,  or  planes,  we 
say  that  two  pencils  of  different 
kinds  are  in  y/iv,sy/c<'///'.'  when 
they  are  made  to  correspond  in 
such  a  manner  that  each  element 
of  one  pencil  lies  in  the  corre- 
sponding element  of  the  other. 
Two  pencils  of  the  same  kind 
are  in  y^r.sy<»v///v  when  each  is 

in    perspective   to    the    same    pencil    of   another   kind.     The    corre- 
spondence   hetween    perspective    figures    is    called    a  prrxpt'cticity. 

\  pencil  of  points  and  one  of  lines  are  therefore  in  perspective 
when  they  lie  as  in  Fig.  5,  where  the  lines  </,  /-,  f,  </,  etc.  correspond 
to  the  points  A,  /.',  (\  I>,  etc.  To  sec  that  we  are  justified  in  calling 
this  relation  a  projeetiyity.  note  that 

AI>       a. I  sin  AOD  _ 
j;i>~    oil  sin  H0l> 

Hence,  if  A  and  />'  are  taken  as  fixed  points  and  J>  as  any  point, 
the  variable  A.  is  a  coordinate  at  the  same  time  of  the  points  of  the 
pencil  of  points  and  of  the  lines 
of  the  pencil  of  lines.  Since  any 
change  of  coordinate  of  either  of 
the  pencils  is  expressed  by  a 
linear  relation,  the  two  pencils 
satisfy  the.  definition  of  projec- 
tive  figures. 

Two  pencils  ('  ranges  )  of  point  s 
are  in  persped  ive  when  t  hev  are 
perspective  to  the  same  pencil 
of  lines  as  in  Fig.  li.  The  st  rai'_dit 
lines  connecting  corresponding 


Y 


<  >  N  E   1  >  1 M  E  N  S  U  >  N  A  L  ( J  E< ) M  ET K  V 


FK.. 


Two  pencils  of  lines  are  in  perspective  when  they  are  in  per- 
spective to  the  same  range  of  points  as  in  Fig.  7.  The  points 
of  intersection  of  corresponding  , 

lines  of  the  two  pencils  then  lie 
on  the. same  straight  line.  'I  hat 
the  relation  is  a  projectivity 
follows  from  I  V,  ^  1  •>. 

From  these  detinitions  the 
following  theorems  are  easily 
proved  : 

7.  If  four  lines  <f  a  pencil  <>f 
linis  are  cut  /<//  ani/  transversal, 
the  cross  ratio  of  the  four  points  of 
intersection  is  independent  of  the 
position  of  (he  transversal  and  is  e>p<al  to  the  cross  ratio  of  the  four  lines. 

II.  If  four  points  of  <t  rani/e  are  connected  icltJi  ani/  center,  the  cross 
ratio  of  t  fie  four  connecting  lines  is  Independent  of  the  position  <f  the 
center  and  Is  eipial  to  the  cross  ratio  of  the  four  points  of  the  ran</e. 

III.  It  the  straight  lines  connecting  three  pairs  of  corresponding  points 
at   tn'o  profeet i ee  rani/es  meet  in  a  point,  all  the  lines  conncctine/  corre- 
xpo/idin;/  points  meet  In  that  point,  and  the  ranges  are  in  perspective. 

IV.  It  the  points  oj  intersection  oj  tliree  pairs  of  correspond iii<i  lines 
ot'  f/i'/i  project  tec  pencils  lie  on  a  straight  line,  the  points  of  intersection 
of  nil  pairs  of  corresponding  lines  lie  on  that  line,  and  the  pencils  are 
in  persj,ec(i,<e. 

The  last  \\\o  theorems  follow  from  III,  vj  15. 

A  pencil  of  lines  is  in  perspective  to  a  pencil  of  planes  when  the 
vertex  of  the  pencil  of  lines  lies  in  the  axis  of  the  pencil  of  planes 
and  each  line  corresponds  to  the  plane  in  which  it  lies.  It  the  plane 
ot  the  pencil  of  lines  is  perpendicular  to  the  axis  of  the  pencil  of 
planes,  the  correspondence  is  a  projectivity,  since,  hv  ^  1  1,  the  same 
coordinate  mav  he  used  for  each  pencil.  If  the  plane  of  the  pencil 
nt  lines  is  not  perpendicular  to  the  axis  of  the  pencil  of  planes,  the 
peneil  ot  lines  is  clearlv  in  perspective  to  another  pencil  ot  lines 
with  its  plum-  so  perpendicular,  for  in  Fig.  7  the  two  pencils  are 
not  necessarily  in  the  same  plane.  Hence  the  relation  here  is  also 

a    projeetU  it  V. 


PROJECTIYITY  23 

EXERCISES 

1.  Consider  ;inv  two  project ivo    pencils  of  lines  not  in   perspective 

and  construct  the  locus  of  the  intersections  of  corresponding  lines. 
Show  that  this  locus  passes  through  the  vertices  of  the  two  pencils  and 
that  it  is  intersected  liv  an  arbitrary  line  in  not  more  than  two  points. 

2.  Consider  anv  two  pencils  of  points  not  in   perspective  and  con- 
struct the  lines   joining  corresponding  points.     These   lines  envelop  a 
curve.    Show  that  not  more  than  two  of  these   lines  pass  through  anv 
arbitrary  point  and  that  the  two  liases  of  the  pencils  belong  to  these  lines. 

3.  Consider  the  locus  of  the  lines  of  intersection   of  corresponding 
planes  of  two  pencils  of  planes  not  in  perspective.    Show  that  this  locus 
contains  the  two  axes  of  the  pencils  and  that  it,  is  cut  bv  anv  arbitrary 
plane  in  a  curve  such  as  is  defined  in  Kx.  1. 

4.  Show  that    if  the  line  connecting  the  vertices  of  two  protective 
pencils  of  lines  is  self-corresponding  (that  is,  considered  as  belonging 
to  one  pencil   it   corresponds   to  itself  considered  as   belonging  to  the 
other  pencil)  the  pencils  are  in  perspective. 

5.  Show  that  if  the  point  of  intersection  of  the  bases  of  two  project! ve 
ranges  is  self-corresponding  (see  Kx.  4)  the  ranges  are  in  perspective. 

6.  (liven  anv  two  protective  ranges  of  points.    Connect  anv  pair  of 
corresponding  points  and  take  anv  two  points  <>  and  <>'  on  the  connect- 
ing line.     With  <>  as  a  center  construct  a  pencil  of  lines  in  perspective 
with  the  lirst  range,  and  with  (>'  as  a  center  construct  a  pencil  of  lines 
in  perspective  with  the  second  range.    Prove  bv  use  of   Kx.  4   that   the 
two  pencils  are  in  perspective.     Hence  show  how  corresponding  points 
of  two  ranges  can  be  found  if  three  pairs  of  corresponding  points  art- 
known  or  assumed. 

7.  Ciiveii  two  project ive  pencils  of  lines.     Take  the  point   of   inter- 
section of  two  corresponding  lines  and  through  it  draw  any  two  lines 
a  and  </.     <>n  n  construct   a  range  of  points  in  perspective  to  the  first 
pencil  ot   lines  and  on  '/'  construct,  a  rair_,re  of  points  in   perspective  to 
the  second  pencil  of  lines.    Prove  bv  use   of   Kx.  T)  that   the  two  ranges 
are  in  perspective.     Hence  show  how  corresponding   lines  of  two  pn>- 
jective  pencils  can  be  found   if  three   pairs  of  corresponding  lines  are 
k nown  or  assumed. 

17.  Other  one-dimensional  extents.    \Vc  have  taken  as  an  example 

ot   a  oiic-ilinieiisioiial  extent  of  points  the  nui"v,  or  pencil,  consist- 

i  .^i 

ingot   all  the  points  on  a  straight  line.     It  is  nhvions,  however,  that 

th;s  i<  nut   the  niilv  example  of  a  one-dimensional  extent   of  [mints. 


lM  UM-:   D1MKNS10NAL  (JKOMKTKV 

In  fact,  unv  curve.  whether  in  the  plane  or  in  space,  is  a  oue- 
dimensional  extent,  the  coordinate  oi  an  element  of  which  may  he 
iletincil  in  a  variety  ot  ways.  (  )ne  ol  the  simplest  methods  is  to 
lake  the  length  «it  the  eim  e  ineiisuix'd  from  a  tixed  point  to  u  vari- 
able point  as  the  coordinate  ot  the  latter  point,  1ml  other  methods 
will  su-_^e>t  themselves  to  the  reader  familiar  with  the  parametric 
representation  of  curves.  In  the  case  of  a  circle,  for  example,  we 
mav  construct  a  pencil  of  lines  with  its  vertex  on  the  circle,  take 
as  the  initial  line  of  the  coordinate  svstem  the  tangent  line  to  the 
circle  through  the  vertex  ot  the  pencil,  and  then  take  as  the  coordi- 
nate of  a  point  on  the  circle  the  coordinate  of  the  line  of  the  pencil 
\\liich  passes  through  that  point. 

Similarly,  the  tangent,  lines  to  a  plane  or  space  curve  form  an 
example  of  a  one-dimensional  extent  of  lines.  Also  the  tangent 
planes  to  a  cone  or  a  cylinder  or  the  osculating  planes  to  a  space 
curve  are  examples  ot  a  one-dimensional  extent  ot  planes.  '1  hese 
extents,  both  of  lines  and  planes,  will  be  discussed  later. 

Moreover,  it  is  not  necessary  that  we  confine  ourselves  to  points, 
lines,  and  planes  as  elements.  We  may,  for  example,  take  the 
circle  in  a  plane  as  the  element,  of  a  plane  <_;vomctrv.  In  that  case 
all  the  circles  which  pass  through  the  same  two  points  form  a  one- 
dimensional  extent,  a  pencil  of  circles.  Another  example  of  a  one- 
dimensional  extent  ol  circles  consists  of  all  circles  whose  centers  lie 
on  a  tixed  curve  and  whose  radii  are  uniquely  determined  bv  the 
positions  of  their  centers. 

In  like  manner  the  sphere  mav  be  taken  as  the  element  ot  a 
>pace  geometry.  All  the  spheres  which  intersect  in  a  tixed  circle 
torm  then  a  one-dimensional  extent  of  spheres,  a  pencil  of  spheres, 
and  other  examples  arc  ivadilv  thought  of. 

In  all  these  cases,  when  the  coordinate  X  of  the  element  of  the 
extent  is  tixed,  the  discussion  of  the  previous  sections  applies. 

<  >ne  more  remark  is  important.  In  all  cases  we  have  allowed  X 
to  take  complex  values.  That  is.  X  is  a  number  of  the  tvpc 


where  /  \  1.  1  he  variable  X  mav  accordingly  be  interpreted  m 
the  u.-ua!  manner  on  the  complex  plane.  The  significance  ot  the 
linear  transformation  mav  then  be  studied  from  the  standpoint  of 


PROJECT!  VITY  25 

the  theorv  of  functions  of  it  complex  variable.  This  lies  completelv 
outside  ot  tlie  ran^e  ol  this  hook. 

We  notice,  however,  that  in  interpreting  X  as  the  coordinate 
of  a  point  on  a  straight  line  we  have  a  one-dimensional  extent  of 
complex  values,  while  in  interpreting  it  as  a  complex  point  on  a 
plane  we  have  a  two-dimensional  extent  of  real  values.  That  is, 
tin-  Jinn  nsi'it*  "f  (in  f.i  ft  /it  :rill  depend  i/]>»/t  H'Jti'tJii'r  it  /.s  cuunt>-d  in 
ttTntx  "/  f"//ij'li-.r  ijuutttiti'fti  "/•  i-f  /•>•<!/  <jt<<titt  tttfx.  \  suallv  we  >hall 
in  this  hook  count  dimensions  in  terms  ui'  quantities  each  of  which 
mav  take  complex  values. 

Consider  the  complex  quantity 


t  being  a  real  quantity  and  tlu-  functions  i'eal  functions. 

Then  as  /  varies,  the  point  X  traces  out  a  curve  on  the  complex 
plane  which  is  one-dimensional.  If  X  is  interpreted  as  the  coordi- 
nate of  a  point  on  a  straight  line,  then  equations  (-)  define  a  one- 
dimensional  extent  of  points  on  the  straight  line,  which  do  not  of 
course  contain  all  the  points  of  the  line.  Such  a  one-dimensional 
extent  of  points  is  called  a  f/t/'f<!'l  of  the  line.  Examples  are  the 
thread  of  real  points  (  \.,  —  (|  ),  the  thread  of  pure  imaginary  points 
(X^-0),  tin.-  thread  of  points  X^  1  +i)  the  square  of  whose 
coordinates  is  pure  imaginary,  and  others  which  can  be  formed 
at  pleasure. 

REFERENCES 


.,  \Vi.NT\vuuni,  ami  SMITH,   Klt'inents  of   I'mjVcti'vo  lioonu-try.    Ginn  and 


Tlicx-  In  M  ik-  difi'rr  fri  'in  t  he  jiivscn!  nnc  in  ln-inu  >yni  In  -tic  in  -lead  i  -I'  analytic  in 
t  ivaTiiirnt.  anil  ilu-y  Lr"  1'i-yuinl  thfCDiilt-iit  nf  unr  I'arl  1  iioli.~cu>.-iiiL:  twu-iliun-n.-ii'iuil 
I'Xtt-nt-,.  In  -pit.'  nt'  ih.i1  thc\-  may  i-asily  In-  n-ail  at  (hi-  jmint.  It'  larirt'i-  ti'rati-rs 
:in-  iircilfil.  ciinsiili  tin-  n-t'nvitc*  -  at  tin-  end  el  1'art  1  1  nl'  this  IMM.U. 


PART    II.     TWO    DIMENSIONAL    (JKOMKTRY 

CHAPTER    IV 

POINT  AND  LINE  COORDINATES  IN  A  PLANE 

18.  Homogeneous  Cartesian  point  coordinates.  Let  < L\'  and  ()Y 
be  two  axes  of  coordinates,  which  we  take  for  convenience  as  rec- 
tangular. Then,  if  /'  is  anv  point  and  /'.)/  is  drawn  perpendicular 
to  <>.\\  meeting  it  at  M,  the  distances  ( > M  and  .)//',  with  the  usual 
conventions  as  to  signs,  are  the  well-known  ('artesian  coordinates 
of  /'.  To  make  the  coordinates  homogeneous  we  place 

"-"-v     •'"'=?•  "' 

Then  to  anv  point  /'  corresponds  a  definite  pair  of  ratios  ./•:  // :  /. 
( 'onversel  v.  to  anv  real  pair  of  rat  ios  ./• ://:',  in  whicli  /  is  not  equal 
to  /.ero,  corresponds  a  real  point.  In  order  that  a  point  mav  cor- 
respond to  any  pair  of  ratios  we  need  to  make  the  following 
definitions,  in  harmony  with  the  general  conventions  of  ^  -\  and  -1  : 

(  1  )  The  ratios  0:0;  0  shall  not  be  allowable,  for  they  make  both 
<>M  and  Ml'  indeterminate,  and  the  point  /'  cannot  be  lixed. 

('!)  ( 'omplex  ratios  shall  be  said  to  represent  an  imaginary 
point  (  ^  ;> ). 

( •)  )  A  set  of  ratios  in  which  f  =  0  shall  be  said  to  represent  a 
point  at  infinity  (sj  I).  In  fact,  it  is  obvious  that  as  /  approaches 
/.ero.  /'  recedes  indefinitely  from  ",  ami  conversely.  In  particular, 
t  he  point  0:1:0  is  t  he  point  at  in  tin  it  v  on  t  he  line  <  >  Y  (  ^  7  ).  t  he 
point  1:0:0  is  the  point  at  infinity  on  t  he  line  "-\ .  and  >/:/»:  0  i> 


19.  The  straight  line.    It   is  a  fundamental  proposition  in  analytic 
ei  nuet  rv  t  hat   an  \    linear  eiiat  ion 


28  TWO   DIMKNSIONAL  (JKOMKTRY 

coordinates  satisfy  an  equation  of  the  form  (1  ),  in  which  the  coetli- 
cit'iits  ure  all  real  and  A  and  /•'  are  not  both  x.ero.  For  proof  of  the 
theorem  we  refer  to  any  textbook  on  analytic  geometry. 

The  proposition  is  a  definition  as  far  as  it  refers  to  imaginary 
points,  to  equations  with  complex  coefficients,  or  to  the  equation 
/•--('.  In  this  sense  "straight  line"  means  simply  the  totality  of 
pairs  of  ratios  ./•://:/  which  satisfy  equation  (  1  ). 

In  particular,  the  equation  f  --=  0  is  satisfied  by  all  points  at 
infinity.  Hence  (ill  point*  «t  hiftniti/  fit-  n>i  a  nfnn';//if  fine,  ailli'il 
!//•>  /hif'  «t  hi  fin  it  i/. 

It  one  or  more  of  the  coefficients  of  (  1  )  are  complex  the  straight 
line  is  said  to  be  imaginary.  It  is  interesting  to  note  that  <ni  hn<t<j- 
inun/  straifiht  tint'  JIKX  i>ne  <nitf  (>»!>/  "»>'  rail  point.  To  proye  this 
let  us  place  in  (1  ) 


Then  (1)  is  satisfied  by  real  yalues  of  .r,   i/,  and  /  when  and  only 

when  . 

a  .r  +  'i  >/  +  <•  f  =  0, 
i  \J         i 

"„•''  +  f'J/  +  ''..'  ==  "• 

These  (Mjuations  have  one  and  only  one  solution  for  the  ratios 
j--.i/:t,  and  the  theorem  is  proved.  Of  course  the  real  point  may 
be  at  infinity. 

Consider  now  any  two  straight  lines,  real  or  imaginary,  with  the 

e(l"ati""S  V  +  /,v/  +  r,  =  0. 

.f.,r  +  //.„;/  +  <'J  =  0. 
These  equations  have  the  unique  solution 


which  represents  the  common  point  of  the  two  lines.  This  point  is 
at  infinity  when  .!,/>'.,  -.•!„#  =  0,  in  which  case,  as  is  shown  in  any 
textbook  on  analytic  geometry,  the  lines,  if  real,  are  parallel.  If 
the  lines  are  imaginary  they  will  be  called  parallel  by  definition. 
\\  e  mav  say 


POINT   AND    LINK  COORDINATES   IN    A    PLANE         -J!) 

If  (.rri,  //n)  is  a  lixt-d  point  on  the  line  (1  ),  we  have 

A(.r- .rif) +  i;(n  ~  nj-    D:  (  i> ) 

II  —  //„  _        - ' 


Whether  J  and   />'  he  real  or  complex  quantities,  there  exists  a 
real  or  imaginary  anjjde  B  such  that 

tan  0=   -4- 
Then,  from  equation  (  L! ). 


P>\  placing  these  equal  ratios  equal  to  >•  we  have,  as  another 
method  of  representing  a  straight  line  analytically,  the  equations 

3'  —  ./•  -f-  /•  cos  0, 
//  =  1 1  -(-  /•  sin  0. 

These  are  the  parametric  equations  of  the  straight  line.  In  them 
./•(i,  y  .  and  0  are  constants  and  /•  a  variable  parameter  to  each  value 
of  which  corresponds  one  and  only  one  point  on  the  line,  and  con- 
versely. It  the  quantities  involved  are  all  real,  the  relation  between 
them  is  easily  represented  bv  a  figure.  In  all  cases 

and  is  defined  as  the  distance  between  the  points  (.r.  >/')  and  (.rn,  _//i ). 
This   work    breaks   down    onlv    when    .  I"  -f-  />'"  =  <>.     In    that    case 
either   .{=/»  =  (),  and   the  line  (1  )   is  the  line  at   infinity,  or  equa- 
tion  (  1  )  takes  the   form 

J'±  "/  +  ''=  °-  <•">) 

I  lere  we  mav  si  ill   place 

tan  0  —  ±  /, 

l)i it  sin  B  ,i>id  cos  0  become  infinite  and  equations  (  -\  )  are  impossible. 
In  fact,  equal  ion  ( '2  )  becomes 

and 

This  shows  that  the  distance  between  anv  two  points  on  the 
imaginary  lines  (.>)  must  be  taken  as  /ero.  For  that  reason  thev 
are  called  mint nni in  fi/ifx.  'I'hev  pla\'  a  unique  and  verv  important 
part  in  the  ^eonu'trv  of  the  plane. 


30  TWO-DIMEXSIOXAL  (JKOMKTKY 

EXERCISES 

1.  Trove  that  through  every  imaginary  point  goes  one  and  only  one 
real  line. 

'2.  Trove  that  if  a  real  straight  line  contains  an  imaginary  point  it 
contains  also  the  con  jugate  imaginary  point  (that  is,  the  point  whose 
coordinates  are  conjugate  imaginary  to  those  of  the  first  point). 

3.  Trove  that  if  a  real  point  lies  on  an  imaginary  line  it  lies  also  on 
the   conjugate    imaginary   line   (that  is,   the    line   whose   eoctlieients  are 
conjugate  imaginary  to  those  of  the  first,  line). 

4.  If  the  usual  formula  for  the  angle  between  two  lines  is  extended 
to  imaginary  lines,  show  that   the  angle  between  a  minimum  line  and 
another  line  is  infinite  and  that,  the  angle  between  two  minimum  lines 
is  indeterminate. 

5.  (liven    a    pencil    of   lines    with    its    yertex    at    the    origin.     Trove 
that    if  the  pencil   is  projected  on   itself  by  rotating  each   line  through 
a  constant   angle,  the  fixed  points  of  the   projection  are  the  minimum 
lines. 

6.  Show  that  a  parametric  form  of  the  equations  of  a  minimum  line  is 

,r  =  .r.  4-  /, 

.'/  =  /',,  ±  ''< 
where  f  is  a  parameter,  not  a  length. 

20.  The  circle  points  at  infinity.  The  circle  is  defined  analyti- 
cally by  the  equation 

a  ( .r  +  >r)  4-  2/./V  4-  -  .<///>  4-  <•/"  =  0,  ( 1  ) 

the  form  to  which  equation  (4).  ^  1'.',  reduces  when  .r.,  _>/n,  and  r 
are  constants  and  ( .r,  // )  are  replaced  by  .r ://:/. 

If  '/  -  0,  the  circle  evidently  meets  the  line  at  infinity  in  the 
two  points  !:/:<>  and  1:  /:<>,  no  matter  what  the  values  of 
the  coefficients  in  its  equation.  These  two  points  are  called  the 
«•//•<•/»'  i>niiitx  at  infinity.  If  // =  0  in  (  1  ),  the  circle  contains  the 
entire  line  at  infinity  and,  in  particular,  the  circle  points.  Hence 
we  may  say  that  <>//  »•//-, •/,•*  //(/.v.v  //*/•«//////  ////•  hrn  <-!r<-l<'  /»'</it* 
'it  in  tin  it  i/. 

I  he  circle  points  1  :  t  / :  0  arc  said  to  be  at  infinity  because  they 
satisfv  the  equation  /  <>.  Their  distance  from  the  center  of  the 


POINT  AND   LINK  COORDINATES   IN   A    IM.ANH        :)  1 

circle  is  not,  however,  infinite.    The  distance  between   two  points 
with  the  nonhomogeneous  coordinates  (y,  >/)  and  (./-,  //  j  is 


»   =  V  /•  -  ro  )-      .//-  .'/„  )-, 
which  can  he  written  in  homoeneous  coordinates  as 


and  this  becomes  indeterminate  when  .r  :  //  :  t  is  replaced  1>\  1  :  _t_-  /  :  0. 

This  perhaps  makes  it  easier  to  understand  the  statement  that 
these  points  lit1  on  all  circles. 

If  ./•  :  //M  :  t>t  is  the  center  of  the  circle  and  r  its  radius,  e<j  nation  (  1  ) 
can  be  written  (compare  equation  (-)) 

(.rtfl  -  ./-,/  )-  4-  (  y0  -  y0t)--r-tit-  =  o. 

When  r=  0  this  e<jnation  becomes 

(j-tn  -  j\t  r  +  (  y0  -  //,/  r  =  o,  c  ;',  > 

the  locus  of  which  mav  be  dt'seribed  as  a  circle  with  center  (.r.  v,  ) 
and  radius  zero.  When  the  center  is  a  real  point  the  ciivle  (o) 
contains  no  other  real  point  and  is  accordingly  often  called  a  pnint 
<•//•<•/!'.  A  point  circle,  however,  contains  other  imaginary  points. 
In  fact,  equation  ('•})  may  be  written  as 

[(./•/.-  j-j  )  4-  i(>/f0—  //,/  )]  [(  •'•',,-  j;f  )  -  i(  //'„-  //  /  )]  =  0, 
which  is  cfjnivulent  to  the  two  linear  equations 
t  (•'•+  /'/)  —  (•'•,  +  /'/  )/  =  °, 


each  of  which  is  satistied  bv  one  of  the  circle  points  at  infmitv. 
Hence  we  have  the  result  that  >/  }»>tnt  <•//•,•/,•  1',,/is/xtx  >>t'  ///,•  in-,, 
imni/iunrif  xfrti/'i//tf  /i/if*  <1r<tirn  fr<»n  tin*  renter  /_•/' //>»•  <•//•-•/,'  /-,  ///,  ///•,, 
cifi'fi'  V"iiifn  iif  inliiiiti/. 

The  distance  from  the  point  (./•,//)  to  anv  point  on  either  of 
the  t\\'o  lines  just  described  is  y.ero,  l>v  virtue  of  etjiiation  ('•*>). 
There  are  therefore  the  minimum  lines  of  >j  1 '.'.  as  is  al>o  directly 
visible  from  ecjuations  (  t  ).  ll  is  olivious  that  through  an\"  point 
tit  the  plane  L^O  two  minimum  lines,  one  to  each  ot  the  cin-le  points 
at  intinitv. 


32  TWO-DIMENSIONAL  GEOMETKY 

EXERCISES 

1.  Show  that  an  imaginary  circle  may  contain  either  no  real  point, 
one  real  point,  or  two  real  points. 

'2.  Consider  the  pencil  of  circles  composed  of  all  circles  through  two 
fixed  points.  Show  that  the  pencil  contains  two  point,  circles  and  one 
circle  consisting  of  a  Mraight  line  and  the  line  at  intiuitv.  Show  also 
that  tin1  point  circles  have  real  centers  when  the  fixed  points  of  the 
pencil  of  circles  are  conjugate  imaginary,  and  that  the  point  circles 
have  imaginary  centers  when  the  ti.xed  points  are  real. 

3.  If  a  pencil  of  circles  consists  of  circles  through  a  fixed  point  and 
tangent  at  that  point  to  a  tixed  line,  "where  are  the  point  circles  and 
t  he  st  might  line  of  t  he  pencil  '.' 

21.    The  conic.    An  ('((nation  of  the  second  decree, 

</.>•-+  '2  Lri/  +  /,//-+  2f.rf  +  -2  </i/t  +  <-'('=  0,  (  1  ) 

represents  a  locus,  called  a  i'<>)ii<;  which  is  intersected  l>v  a  general 
straight  line  in  two  points.  For  the  simultaneous  solution  of  the 
('((nation  (  1  )  and  the  ('((nation 

A.r  +  liil  +  <'t  .     i>  (-J) 

consists  of  two  sets  of  ratios  except  tor  particular  values  of  .1,  /•', 
and  ('. 

Let  toe  ('((nation  (1  )  be  written  in  the  nonhomogeneous  form 
bv  placing  /—  1,  and  let  ( '2 )  be  written  in  the  form  (^19) 

.r  =  .?•  +  r  ci  >s  0,  if  =  ij(t  -f-  r  sin  0.  ('-\ ) 

The  values  of  /•  which  correspond  to  the  points  of  intersection 
of  the  straight  line  ('!)  with  the  curve  (1)  will  be  found  by  sub- 
stituting in  (  1  )  the  values  of  .r  and  //  j^ivcn  by  (  :>> ).  There  results 

Lr+  2  .!//•  +  .V-  0,  (^4) 

where  .17  =  (".'•„+  /'//,,+.Or(ls  ^  +  ^"',,+  ^/,,4-//)sin  ^. 

This  \\  ill  be  /.en*  for  all  values  of  0  when  .r<t  and  //  satisfy  the, 
equations  'Wn+ 7///u+/=  0,  //./;, 4-  /'.'/„  +  //=0.  (.">) 

In  tin-  case  the  point  (./;.//)  will  be  called  the  I'^nicr  of  the 
curve,  since  anv  line  through  it  meets  the  cniA'c  in  two  points 
•  •ijuallv  distant  from  it  and  on  opposite  sides  ot  it.  Now  c(|iiation 
(•>)  ran  lie  satisfied  hv  a  point  not  on  the  line  at  iulinitv  \\'hen 
ainl  olilv  when  //-  -  nl,  --  0.  Hence  ///«•  cnttii'  (]  )  ?'x  n  t'r>it,-'i/  i-»/i/<' 
i/'/n  //  //"  -  lit,  i=.  II,  dm I  /.s  //  iiniti'i'ilt ml  ('nil!''  H'lii'il  //" —  nfi  --  <l. 


POINT   AND    LINK  COORDINATES    IN    A    PLANK         ;J:J 

The  conic  (1  )  is  cut  by  the  line  at  infinity  /  =  <)  in  t\v<>  points 
for  which  the  ratio  ./•://  is  g-iveii  by  the  equation 

(U--+  -  //./•//  4-  ////-  —  0.  ( ii ) 

This  has  equal  or  unequal  roots  according  as  Ir — <iJ>  is  equal  or 
unequal  to  /ero.  Hence  <i  <•<  ntr<il  <•"///<•  t-i/fx  tin'  lnn>  *it  t'ufinif//  in  f//'» 
ilfx//tti-f  pnintx  ;  ii  lio/u't'itt rill  cvin<:  <'utx  tin'  Inn'  «t  tnfiniti/  in  fim 
('"hn'iili'iit  i>i>!ntx, 

So  jar  the  discussion  is  independent  of  the  nature  of  the  coel'li- 
cicnts  of  (1  ).  If.  however,  the  coefficients  are  real  the  classifica- 
tion mav  be  made  more  eloselv,  as  follows: 

(1)  h2  —  ab<0.  The  cum;  cuts  the  line  at  infinity  in  t\vo  distinct 
imaginary  points.  I;  is  an  ellipse  in  the  elenienlarv  sense,  or 
consists  of  t\vo  imaginary  straight  lines  intersecting  in  a  real 
point  not  at  intinitv,  or  is  satisfied  bv  no  real  point. 

(!')  h2  —  ab>0.  The  curve  cuts  the.  line  at  infinity  in  two  dist  inet  real 
points.  Tt  is  a  hyperbola  or  consists  of  t  \vo  real  nonparallel  lines. 

(',})  h2  —  ab  =  0.  The  curve  cuts  the  line  at  infinity  in  two  ]'cal  coin- 
cident jioints.  It  is  a  jiarabola,  or  two  parallel  lines,  or  two 
coincident  lines.  In  the  very  special  case  in  which  /,=</  =  //  —  0 
it  degenerates  into  the  line  at  infinity,  and  the  straight  line 
j;r  +  ,,,,  +  ,,  -  o. 

EXERCISES 

1.  Show   that    for  a   given   conic   there   goes    through   any    point,    in 
general,  one  Mrai^ht  line  such  t  hat  1  he  segment  intercepted  by  the  conic 

is   1  liseeted    bv    t  lie    poilll  . 

2.  Show  that  for  a  given  conic  there  go  through  any  point,  in  gen- 
eral, two  lines  which  have  one  intercept   with  the  conic  at.  infmitv. 

3.  Prove   that   through   the   center   of  a   central    conic   there  go  two 
straight  lines    which    have   both    intercepts    with   the   conic   at    intinitv. 
These  are  the  «.ii/ni />t"fi'x.    Show  that    the  iisvmptotes  of  an  ellipse  are 
imaginary  and  Ilio-i-  of  a  hyperbola  real,  and  find  iheir  equations. 

4.  Show  from  i  .">  i  that    if  ./•.:  i/n:  t    is  a   point   on  the  conic,  the  equa- 
tion of  the  tangent    line  is 

(ii.r  :  -f-    'a  ij    +,/'/.,)  .''  -\-  i  A.'1,,  +  /<//,  +  ,'//,,)  .'/  +  (  /''•    -\    .'/.'/,  4-  '''    >  f         ()- 

5.  Show  that  the  condition  that   i  1  )  should  represent   straight  lines  is 

n       'a     f 

l>        '/     =  0. 


34 


TWO   Pl.MKNSLONAL  (JKO 


22.  Trilinear  point  coordinates.  Let  .!/•',  /•'(',  and  ('A  (Fig.  8) 
be  three  fixed  straight  lines  of  reference  forming  a  triangle  and  let 
/.- ,  /r.,,  and  /^  he  three  arbitrarily  assumed  constants.  Let  /'  be  anv 
point  in  the  plane  AIK'  and  let  /<r  /',,  and  y>.t  be  the  three  perpen- 
dicular distances  from  /'  to  the  three  lines  of  reference.  Algebraic 
si<^ns  are  to  lie  attached  to  each  oi  these  distances  according  to 
the  side  of  the  line  of  reference  on  which  /'  lies,  the  positive  side 
of  each  line  being  assumed  at 
pleasure. 

The  coordinates  of  /'  arc 
defined  as  the  ratios  of  three 
quantities  ./•  ,  .>'.,,  ./•„  such  that 

It  is  evident  that  if  /'  is  given, 
its  coordinates  arc  uniquely  de- 
termined. Conversely,  let  real 
rat  ios  (i  :  </„ :  ti  be  assumed  for 
.r  :./•„:./-.  The  ratio  j-  '.rn  =  a  \<ia 

furnishes  the  condition  —  =  con- 
stant, which  is  satisfied  hy  anv 
point  on  a  unique  line  through  A.  Similarly,  the  rat  io  .r, :  .?'„  ~ '*., :  ". 
is  satisfied  by  any  point  on  a  unique  line  through  ( '.  If  these  lines 
intersect,  the  point  of  intersection  is  /',  which  is  thus  uniquely 
determined  by  its  coordinates. 

In  case  these  two  lines  are  parallel  we  mav  extend  our  coordi- 
nate system  by  saving  that  the  coordinates  <t  :  ",:  <'.(  define  a  point 
of  infinity.  These  are,  in  fact,  the  limiting  ratios  approached  by 
j-{  :,'•.:./•„  as  /'  recedes  indefinitely  from  the  lines  of  reference. 

We  complete  the  definition  of  the  coordinates  by  saving  that 
complex  coordinates  define  imaginary  points  of  the  plane,  and  the 
coordinates  0  :():<>  arc  not  allowable. 

The  coordinates  of  .1  are  <):<>:  1,  those  of  /•'  are  0  ;  1  :  0,  and 
t  IMKC  of  ('  arc  1  :  <>  :  ".  The  ratios  of  /,-  .  /-  ,  and  /,'  are  determined 

I  J  :! 

when  the  point  with  the  coordinates  1  :  1  :  1  is  fixed.  This  point  we 
shall  call  the  unit  point,  and  since  the  /•'*  are  arbitrary  il  mav  lie 
taken  anywhere.  Ilence  the  r<,///v////^/V  \//.s-/V///  /'a  J,  /<•/•////'//<•</  /<//  tJim- 
<t rl:/f i-ii r*i  ////<x  '//  ri'fi'1'i'Hi'i  <tu<l  <'n  ttrl>it rnry  unit  pmnt. 


POINT  AND   LINK   COORDINATES   IN    A    1'LANK         ;;:, 

The  trilinear  coordinates  contain  the  ('artesian  coordinates  as  a 
special  limiting  ca>e,  in  which  the  line  IK'  is  the  line  at  infinity,  if 
IK'  recedes  indefinitely  tnun  K  /',  becomes  infinite,  but  the  factor 
/•(  can  be  made  to  approach  /ero  in  such  a  way  that  L'nn  /c,ji,  —  \. 
(  'fhere  is  an  except  ion  only 
when  /'  is  on  the  line  IK  '  and 
remains  there  as  IK'  becomes 
the  line  at  intinit  v  :  in  this 
ease  /•./'.---  ().  )  It'  in  addition 
we  place  /,'  -  /,',  1,  the  coi'ir- 
dinates  ./•  :  .i\t :  j\  beet  une  the 
coi'irdinat es  ./• :  // :  t  of  i;  Is. 

23.  Points    on    a    line.       If 
//  :  //, :  _//.,  and  2  :  ,?, :  2.,  <tr<'  tn'<> 
ti.fii/  i>"intfi,    tin'  rn,",i',l!n<iti'x  i>t' 
an  if    ji'ilnt    nit    tin1    tttntiifht    laic    "^V-1 
Jdiniiit/   tJiP.ni   «/'>'   if  -\-  X.r  :  //„ + 

X,:,:   //..  -f-  X.;..,    (tml   tin//  />"/'//(    irt'f/i    tltt'M'    ruo/'iUtnitt'S    //Vx    <>n    tJlftt    I! in'. 
To   prove  this   let   )'  and  /  (  l-'i^.  (J )  be  the  two  fixed  points  and   1' 

anv  point  on   the  straight    line    YZ.     Place       ,   --;/;.    Then,  if  />[,  /',, 

and   i^  are    the   perpendiculars   from    ),    /',   and    Z  respectively  on 
A  //,  it  is  e\  idciit   Irom  similar  triangles  that 

/',       /','    ' 

r"  -/', 
r\  + 


FIG.  U 


\vlielice 
Similarly, 


\\hcre   p.   p' ,    and    p"    are    proportionality    tador-.      l'.\ 
\\  •(•    ha\e 


30  TWO  -DIMENSIONAL  GEOMETRY 

The  above  proof  holds  for  anv  real  point  /'.  Conversely,  any 
real  value  of  X  determines  a  real  ///  (the  coordinates  of  1"  and  Z, 
liein^  real  )  and  hence  deteniiines  a  real  point  of  7'.  For  complex 
values  of  X.  or  for  imaginary  points  )"  and  '/,  the  statement  at  the 
beginning1  of  this  section  is  the  dt'finiiton  of  a  straight  line. 

It  is  to  lie  noticed  that  X  is  an  example  of  the  kind  of  coordinates 
ol  thi'  jxiints  of  a  ran^'e  which  \\'as  discussed  in  sj  S. 

24.  The  linear  equation  in  point  coordinates.  A  humvyeneouts 
t  ijiitittt'n  'if  the  tit'xt  (/(•///*<'(', 


/'i  ju't  -xi'  nt  x  it  xti'tntfht  ///if,  i/inf  ('(iiti'O'Nctj/. 

To   pro\  e    this   theorem   it    is  necessary   to  sliow  that    the    linear 
equation  is  equivalent  to  the  equations  of  ;<  •2'-}.    Let  us  have  pven 


I-'roiu  these  three  equations  we  have 


Then   from   the   theorv  of  determinants  tin-re  exist   three   multi- 
pliers Xf   X,.  X,  such  that 


p.'',—  ,'/    \- 


J'OINT  AM)    LINK  COORDINATES   IN    A    1'LANK         :J7 

The  elimination  of  p  and  X  then  i;'ives 


\\'liich  is  11  linear  equation  in  ./-f  ./•„.  and  ./•.. 

Hence    equation    (1)    is    equivalent    to    equation    (  L'  ),    and    the 
theorem   at    the   beginning  of   this  section    is  proved. 

25.  Lines  of  a  pencil.  //' 


It  is  evident  tliat  (  •>  )  re[»resents  a  straight  line  and  thai  the 
eoi'irdinates  of  anv  point  uhieh  satisfy  (1  )  and  ('2)  satisfy  also  (  :\  ). 

Furthei'iuore,  X  is  uniquely  determined  l»v  the  eoiirdinates  of  anv 
point  not  on  (1  )  and  (-).  lleiiee  for  all  values  of  X,  (  ;>  )  delines 
the  lines  of  a  pencil. 

The  parameter  X  in  (  o  )  is  of  the  tvpe  of  eoi'irdinates  detined  in 
vj  1".  To  show  this  let  us  take  )'(//  ://.,://.),  a  point  on  (1),  and 
'/.  (  ~.  :v,:.r,),  a  point  on  ('-}.  Then  //  +  X^  :  //.,+  Xr,  :  i/  ,  -f-  X.r,  is  a 
point  on  ('-}  )  and  al>o  a  point  ot  the  ratine  determined  1»\"  )  and  /.. 
l>v  ^  '.'.  X  is  the  eool'dimite  ol  a  point  on  the  raiiijv.  and  heiiee,  as 
shn\\M  in  ^  l»i,  the  eoi'irdinate  of  a  line  of  the  peiieil  in  the  sense 
of  ^  1<». 

EXERCISES 

1.  Show  thai  the  equation  of  anv  line  through  the  point  .1  ot'  the 
tnaii.uh1  of  re  fere  net'  is  .''-}-  A./1.,  =  0,  and  tind  the  etMU'dinates  of  the 
jioint  in  wliich  it  inter>ret  s  aii\  line  <t  ./•  -f-  ".,.'•.,+  "  .-''.  -  :  *'•  Distinguish 

hi-t\Vrr!l     till1     eases     ill      \\llirll     II       :-    (I    alnl     II,        -    0. 

','..  \\'ril>'  tin-  erjuatioiis  <i|'  two  prujrct  ive  ]M-nrils  of  liin-s  with 
the  Veil  ices  I  and  /.'  respect  ivrlv.  l''ilnl  the  cijiialinn  >ati-tird  hv 
thr  roordiiialt-s  of  tin-  points  of  inirr>rrt  ion  of  corresponding  hnrs. 
Hence  verif  I'lx.  1.  ?  1C. 


38  TWO-DIMENSIONAL  GEOMETRY 

4.   Show  that  homogeneous  point  coordinates  are  connected  by  the 

relation 

P  ("/••/,  +  Mya  +  t'kgf,)  =  A  , 

when-  ",  I',  and  '•  arc  tlie  lengths  of  the  sides  of  the  triangle  of  reference 
and  A"  is  its  area.    Hence  show  that 


is  the  equation  of  tlie  straight  line  at  infinity. 

5.  ('(insider  the  case  in  which  I!  is  at  infinity,  .1  and  <'  are  right 
angles,  and  /,',  =  /.-.,=  />-;)  =  1.  Show,  for  example,  that  ./^  -f  j-.(  ~  0  is 
the  equation  of  the  straight  line  at  infinity  and  that  ./^  +  ./^  -f-  Ar._,  -—  0 
is  the  equation  of  any  straight  line  jiarallel  to  .If. 

26.  Line  coordinates  in  a.  plane.  The  coefficients  a},  </.,,  '/.,  in  the 
equation  of  a  straight  line  are  sullir.ient  to  fix  the  line.  In  fact, 
to  anv  set  of  ratios  ^  :  </a  :  «3  corresponds  one  and  onlv  one  line, 
and  conversely.  These  ratios  may  accordingly  lie  taken  as  coor- 
dinates of  a  straight  line,  or  lint-  cwnUnatt's,  and  a  geometry  may 
he  hu  ilt  up  in  which  the  element  is  the  straight  line  and  not 
the  point. 

A  variable  or  general  set  of  line  coordinates  we  shall  denote  bv 
it  :  H  :  n  .  and  the  line  with  these  coordinates  is  the  straight  line 

1          •_'          :;  O 

which   has   the   point   equation 

Vi+  "-''••+  'V'a^0'  0) 

This  equation  may  also  be  considered  as  the  necessary  and  suffi- 
cient condition  that  the  line  i^  :  (/.,  :  n^  and  the  point  J\  '•?„'•  f.j  are 
"  united  "  :  that  is,  that  the  point  lies  on  the  line  and  the  line 
passes  through  the  point. 

It  is  oh\  ions  that  the  definition  of  line  coordinates  holds  for 
('artesian  as  well  as  for  trilinear  coordinates.  With  the  use  of 
trilinear  coordinates  any  straight  line  may  be  given  the  coordinates 
1:1:1.  For  the  substitution 


\\hirh  amounts  to  u  change  in  the  constants  /-  .  /-|(  /-  in  (  1  ), 
i  •_!'.!.  changes  the  etjuation  n^  >  ^  +</...'.,  f  '',-'',—  (J  into  the  equation 
./•!  -  ./•'  •  ./  '  -  n. 


POINT   AND    LINK  COORDINATES    IN    A    I'l.ANK         -J'.» 

27.  Pencil  of  lines  and  the  linear  equation  in  line  coordinates.     If 

r  :  i'  :  >•    mid  u1  :  <"'.,:  ir^  are  two  fixed   lines,  it    follows   inuuediatelv 

f'niiii  £  _•">  that 

t'i+  X//^  :  /-.,-f  X"1,  :  ''.,-f-  X"';!  (  1  ) 

represents  anv   line  ot    tlu-  pencil  determined    l>v   tin-   tun    lines  rt 
ami  n\. 

('(insider  now  an  equation  of  the  first  decree  in  line  coordinates, 


It  may  lie  readilv  shown,  as  in  ^  '24,  that  it  /•  :  r  :  r.  and  //•  :  //',  :  >r  ^ 
are  two  sets  of  coordinates  satisfying  (  -  ),  the  general  values  of 
//  :  ".,  :  "j  which  satisfv  (  '2  )  an-  of  the  form  (1  ).  Ileiiee(-l)  repre- 
sents a  pencil  ol  lines. 

<  >r  \\  c  mav  ai''_;Mie  dircctlv  from  (  1  ),  ^  iM,  and  sav  at  once  that 
anv  line  \\hose  coordinates  satisfv  ('2)  is  united  with  the  point 
<i  :  ti  ::  n,  and,  conversely,  that  anv  line  united  wit  h  the  point  <i  :  ti  t:  <i 
has  coi'u'di  nates  which  satis!  v  (  -  ).  \\  e  have,  therefore,  the  theorem  : 

77/c     t'/l/<lf/n/t      ,1/1    -)-   (/    tl  ^  -(-  <(    II  _--   0     rt'Jt/'fXi'tttx     It    jit'/lr/l     <>t    ll/lfX     <>t 

,'/•//  /••//  I/if  fcrfi-j.-  t'x  ///i'  point  ii   :  iit:  <i:. 

Compare  the  linear  ecjuatioii  in  point  coordinates, 

<V'|  "+"  ''..'''.+  "»•'';(  =   ()'  ('*>  > 

and   the  linear  eipiat  ion   in   line  coordinates, 

ii  n  -)-  ".,//.,+  <t.n.=  ".  (  1  ) 

[Cipnitiou  (  :>>  )  is  satisfied  l>y  all  points  on  a  I'an^e  of  which  t  he 
base  is  the  line  witli  the  line  coordinates  <i  :  '/,:  n..  It  is  the  />"i//t 
1'ijtiiit  /»//  nt  llnit  Inn'. 

M(|iiat  ion  (1)  is  satisfied  bv  all  lines  of  a  pencil  of  wiiich  the 
vertex  i--  the  point  with  the  point  coordinates  it  :  dt:  ti  .  It  is 
the  It  n<-  1'ijliafi'iH  "f  thill  jxi/'/it. 

EXERCISES 


40  TWO   D1.MKNS10NAL  (iEOMETKY 

3.  Find  in  line  coordinates  the  equations  of  tlie  points  of  the  range 
which  lie  on  tin1  line   1:1:1;   also  the  point  coordinates  of  the  same 
range. 

4.  Find  in  point  coordinates  the  equations  of  tlie  lines  of  the,  pencil 
\vith  vertex  1:1:1.    F'ind  also  the  line  coordinates  of  the  lines  of  the 
same  pencil. 

5.  Show  that   line  coordinates  are  proportional  to  the  segments  cut 
nff  l>v  the  line  on  the  sides   of  the  triangle  of  rctereiice,  each  segment 
hciiiLT  multiplied  l>v  a  constant,  factor. 

(1.  Show  that  line  coordinates  are  proportional  to  the  three  perpen- 
diculars from  the  vertices  of  the  triangle  of  reference  to  the  straight 
line,  each  perpendicular  being  multiplied  liv  a  constant  i'actor. 

28.  Dualistic  relations.  The  geometries  of  the  point  and  the  line 
in  a  plane  are  dualistic  (  £  -  ).  This  arises  from  tin;  fact  that  the 
algebraic  analysis  is  the  same  in  the  two  geometries.  The  differ- 
ence conies  in  the  interpretation  of  the  analysis.  In  both  cases  we 
have  the  two  independent  ratios  of  three  variables  which  arc  used 
homogeneously.  In  the  one  case  these  ratios  are  interpreted  as  tlie 
coordinates  of  a  point;  in  the  other  case  they  are  interpreted  as 
the  coordinates  of  a  line.  In  both  eases  we  have  to  consider  a 
linear  homogeneous  equation  connecting  the  variables  which  is  sat- 
istied  bv  a  singly  infinite  set  of  ratio  pairs.  In  the  point  geometry 
this  equation  is  satisfied  by  the  singly  infinite  set  of  points  which 
lie  on  a  straight  line.  In  the  line  geometry  this  equation  is  satis- 
lied  bv  the  singly  infinite  set  of  straight  lines  which  pass  through 

t  O     t.  O  o 

a   point. 

From  the  above,  it  appears  that  any  piece  of  analv.Ms  involving 
two  independent  variables  connected  by  one  or  more  homogeneous 
linear  equations  has  two  interpretations  which  differ  in  that  "  line  " 
in  one  is  '  point  '  in  the  other,  and  vice  versa.  Hence  a  geometric 
theorem  involving  points  and  lines  and  their  mutual  relations  mav 
be  changed  into  a  new  theorem  by  changing  point  to  line  and 
"line"  to  "point."  In  making  this  interchange,  of  course,  such 
other  changes  in  phraseology  as  will  preserve  the  FOnglish  idiom 
are  also  necessary.  For  example,  "  point  on  a  line  "  becomes  u  line 
through  a  point,"  and  "a  line  connecting  two  points  "  becomes  "  a 
Mi'mt  of  intersection  of  two  lines." 


POINT  AND    LINE  COORDINATES    IN    A    PLANK        -11 


We    restate    some    of    the    results   thus    far   obtained    in    parallel 
columns   so   as   to   show   the   duulistic   relations. 

The  ratios  s  :  j\,;  s  determine  The  ratios  it  :  nml:  H  determine 

a  point.  a  straight  line. 

A  linear  equation  ",.'', -f- ".,•''.,-)-  A  linear  equation  ",", -f- ".,".,  4- 

n  ..r,  --  0  represents  all  points  on  «?/==  0  represents  all  lines  through 

the  line  of  which  the  coordinates  the  point  of  which  t  he  coordinate-, 

are"  :".,:"..  It  is  the  equal  ion  of  are  a  :"„:",.  It  is  t  he  eijuatiuli  of 

the  line.  the  point. 

If  _//,  and  ,-;,  are  fixed  points  the  If  c,  and  t/\  are  tixed  lines  the 

coordinates  of  anv  point  on  the  coordinates ofan v  linetliroui^h their 

line  connecting  them  are  i/t  -\-  Av,.  point  of  intersection  are  /•,.  -f  Ac1,. 

If     nfrl+":fr,.+  li.Jft=0 

and       ''/!  +  Va  +  Vs  =  ° 

are  the  eipiations  of  two  lines,  the  are  the  eipiations  of  two  points, 
equation  of  any  line  through  their  the  equation  of  anv  point  on  the 
point  of  intersection  is  line  connecting  them  is 

,/  „    4-  „  ,/    4.  „    ,/ 

11'          -i     '2     '  :;       :i 


Three     lilies     C,,     /'',.     (i t    meet    ill 

a   point    when 


Three  st  raight  lines 

V-/  ,-  =  (i,  V/, .,-  —  o,  V,.  ,.  =  o 

<L*    •  —    '  ^-   • 

meet    in  a  point   \\  hen 


Three    point' 
V",/      HI. 


0 

' 


'  ^  ' 

lie  on  a  st  raight   line  u  hen 

"       /, 
i        i        i 

II 


29.  Change  of  coordinates.  \\Y  \\ill  tir>t  estalili-h  the  relation 
lietweeil  a  set  o|  ( 'artesian  ci  li'inlinat  es  and  a  set  ot  t  riliin-ai'  eoiir- 
dinates.  Let  .11',.  li(\  and  ('A  l»e  the  lines  of  reference  of  the 


4*2  TWn-lUMKNSloNAL  (i  K(  ).M  KTK  V 

triliiu'iir  coordinates  and  let  their  equations  referred  to  anv  set  of 
Cartesian  coordinates  lie  respectively 


Then  l>v  a  familiar  theorem  in  analytic  geometry 
/'i=— ^  tt'f    £-> 

_  "2-''  +  '''•j//  +  <'J 

1   •! 


\\"e  mav  take  without   loss  of     eneralit 


since    each    of    the    eijuations    (1)    may   l>e    multiplied    l>v    ;i    factor 
\\ithoiit   chan^in^  the   lines  represented. 

Therefore  \ve  have 

pj'l=tilj'  +  l,l,/  +  ,']t, 

PJ;,=  ti^'  +  ljj  4-  ••./,  (;J) 

p.r.  =  <i  ..r  +  l.ji  +  <-.t, 

where  p  is  a  proportionality  factor. 

Since  the  lines  A  //,  JK\  and  '".i  form  a  triangle,  the  determinant 
'//',/•,  does  not  vanish  and  equations  (  -  )  mav  be  solved  tor  j\  //, 
and  /. 

Suppose  now  another  triangle   A'/t'C'  lie  taken,  the  (Mpiations  of 

its   sides    l)eillL,r 


and    let    ./'|  :./•.',:./•'.   lie   trilinear   eoi'u'dinales   referi'ed    to  the   triangle 
A'  !>'<  ''.     Then,  as  before, 

p'-''i  -  "[•'•  4-  '>[</  4-  -•;/, 

f/./  '       "'•'    f   /-.I'/  4-  ''.V,  (  >  ) 

//./•',  -    '/'./•  f  A.',//  f  '•'/. 


POINT   AND    LINK  <  'OuKl  >IN  ATKS    IX   A    PLANK 


Equations  (-)    niiiy   IK-  solved    for  ./•,  //,   and   t   and   tlu1   results 
substituted  in  (4).     There  result   relations  ot   the  form 


n  .i 
cr.r 


/Vi  +  /l.r,  +  #,./•.„  (;)) 

7,.r,+  7rr-+  7:l.»-:1, 

which    are    the    equations   of    transformation    of    coordinates   from 
.r,  :  .r.,' :  .r,  t  < »  r\  :  ./'',  :  .r', . 

In  (•"">)  the  ri^ht-hand  members  e(juated  to  /ero  jjjive  the  eijiia- 
tions  in  trilinear  coi)rdinates  of  the  sides  of  the  triangle  of  reference 
A' !'>' < '' .  Since  these  do  not  meet  in  a  point  the  coefficients  are  sub- 
ject to  the  condition  that  their  determinant  does  not  vanish,  and 
this  is  the  only  condition  imposed  upon  them. 

P>v  the  transformation  (•"">)  the  equation  of  the  straight  line 


becomes  \i\.r\  -f-  //.',.'•'  -j-  n'^'-i  —  0, 

where  pit{  =  al><(  +(3^i',  -f  7i",^ 

P  "  J  =    "  ,-  "  l'   +  fi.2  ».J    4-    7..  "  ';  ,  (    '  >    ) 

^".i   =   "V'l   +/^;i"'    +    7;;"';- 

These  are  the   formulas  for  the  change  of  line  coordinates. 

In  connection  with  the  change  of  coordinates  three  theorems  are 
of   importance. 

7.   T/II'  ili'ijfi'i'  nf  iin  i^jimt  inn  i  ii  tmlnt  in'  //'iii'  I'niirtl  /mite*  ix  unnltcred 
In/  it  rJuDii/i'  t'rn/n  "Hi1  ,s'*7  >,('  t  ril  i  in-ur  I'm'iri/itKift'H  /"  <1)lt>fht')'. 

II.  If  tJit-  cnfir<?innti'K  //.  <iml  zi  <tr>'  trnn>tfi>r)m'<l  i»f"  (I"'  i'fn'h'i1in<it?8 
//'  n  ml  ,r',  (/i  f  I'nfli'iUtHiti'x  v,  +  X.;.  '//v  t  r<tiist'i>ri»>  <l  inf<>  (//>'  ewnlintid'H 

>/'•    +    X'z|.,     H'Jlt'n1    \'  i'\,     I'    i'»  /'//</    it    I'alixtilllt. 

III.  Tin'  -r«xx  riifi'i  >>f'    'nitr    x'i/ifx  <>r    »ur  //;/<>•  in  uiiJrcnJi'iit  <>f 


Theorem  I  follows  immediately  from  the  fact   that  e<juations  (^;> 

and  (  i!  )  are  linear. 

To    prove    theorem    II    imte    that     from    (  •">  ),    it'    the    coordinate 
.'/,-4-  X-~,  are  transformed  into  ./•',  then 

a-.r[  =  n}(>/^  +  \.^  )  +  a.,  (  //.,  +  Xr.  )+'(,(  //.  +  X,r.  ) 
—  (  a^  4-  a,  ,//.,+  'i.,//.  )+  A  ('(,:,  4-  '(./,,  4-  '(  ,.::.  ) 


T\V( )-  DIM  KN S10N  AL  C  K(  >M  KTK  V 


where  (T  and  cr ,  are  used,  since  in  transforming  //.and  z{  l>v  (_;">)  (lie 
proportionality  factors  mnv  differ. 

Similar  expressions  mav  be  found  for  X  and  ./•',.     Hence  \ve  have 

ii         i&ii^-ii^'t          i  •  i 
./-,  :  ./•.,  :  .r,  -=  >/,  +     ~  \~,  :  //.,  4-  •-  X.?.,  :  //..  +  -  "  A.^,     winch     proves     the 

°"i  /, 

theorem.  The  same  proof  holds  for  line  coordinates  using  equa- 
tions (•',). 

Theorem   III  follows  at  once  from  II. 

30.  Certain  straight-line  configurations.  A  wmpJctr  n-?hh-  is 
defined  as  the  figure  formed  by  ti  straight  lines,  no  three  of  which 
pass  through  the  same  point,  together 
with  the  .',  )i  (  n  —  1 )  points  of  inter- 
section of  these  lines.  A  complete 
three-line  is  therefore  a  triangle  con- 
sisting of  three  sides  and  three  vertices. 
A  complete  four-line  is  called  a  com- 
plete quadrilateral  and  consists  of  four 
sides  and  six  vertices.  Thus  in  Fig.  10 
the  four  sides  are  ".  f>.  <-,  </  and  the  six 
vertices  are  A'.  /,,  .)/,  -V,  /',  (t>.  Two 

vertices  not  on  the  same  side  are  called  opposite,  as  A"  and  .17,  L 
and  .V.  /'  and  (,>.  A  straight  line  joining  two  opposite  vertices  is  a 
diagonal  h'tif.  The  complete  quadrilateral  has  three  diagonal  lines. 

A  I'ompJetc  n-p'iint  is  de- 
fined as  the  figure  formed  by 
n  points,  no  three  of  which  lie 
on  a  straight  line,  together 
with  the  \»(r>—  1)  straight 
lines  joining  these  points.  A 
complete  three-point  is  there- 
fore a  triangle  consisting  of 
three  vertices  and  three  sides. 
A  complete  four-point  is  called 
a  complete  quadrangle  and 
consists  of  four  vertices  and 
six  sides.  Thus  in  Fig.  11  the  four  vertices  are  . I.  /•'.  ('.  />  and 
the  six  sides  are  k,  /,  m,  n,  p,  q.  Two  sides  not  passing  through  the 
same  vertex  are  called  opposite,  as  k  and  ?/>,  I  and  n,  and  /<  and  y. 


POINT  AND  LINK  COORDINATES  IN   A    PLANK        4 .1 


The  point  of  intersection  of  two  opposite  sides  is  a  diagonal 
point.  The  complete  quadrangle  has  three  diagonal  points. 

It  is  obvious  that  a  complete  /(-point  and  a  complete  //-line  are 
dualistie.  A  triangle  is  dualistie  to  a  triangle,  and  a  complete 
quadrangle  to  a  complete  quadrilateral.  '1  lie  diagonal  lines  of 
a  complete  quadrilateral  are  dualistie  to  the  diagonal  points  of  a 
complete  quadrangle. 

For  the  complete  triangle  we  shall  prove  the  following  dualistie 
theorems  : 

7.  The  theorem  of  Desargues.  If  t>r<>  //•/// /////»•*  are  *»  y//^v<7  tint  f/n> 
stt'itfi/Iit  If'it'-x  i'"ii  ni'i'ti  inj  /I'l/it'i/ni/i/iix  >•*'/•( ii'fx  ID,  ,-t  In  /(  ji'>inf,  t/n'ii  ///•; 

l)0//ltX     lit     /  lltl'/'Xl'i'f/'i/l     "t      Iln/lll'lni/illIX     ft/i/l  'X     III'     "I!     it     xfrnil/ltt     Jill,'. 

II.      If     I"''!       f,-iil/ll//l'X      III'!'      X'l      jl/llri',/      tl/'lf      till'     Jln/'ll/K      nf     llltl  '/'X,  'i'f/'"/l 

nf    Jiu//i'i/",/iilix    */'</<  x    //i'    nil     il    sf/-il/f///f    l//ii',     tin'))    tin'    f/'/n'X    i'"iin, 'I'tiiliJ 

Let  there  be  given  two  triangles  with  the  vertices  J,  /•',  ('  and 
J',  /'',  (''  respectively  (Fig.  ll')  and  with  the  sides  //.  n,  /•  and 
u\  //',  '•'  respectively,  the 
side  '/  Iving  opposite  the 
Vertex  J  etc. 

\\'e  shall  denote  bv 
A  A'  the  straight  line 
connecting  A  and  J', 
and  bv  a<t'  the  point 
i  if  intersect  n  m  ot  it  and 
'/'.  Then  the  two  the- 
orems stated  above  are 
respect  ivelv  : 

If    tin'     Kf  /•'//;////      1 1 /I  I'* 

A.I',  /:/:',  <>/i<l  rr'  M,-,-t 

In   it   point  0,   tin'  jiaitttx  nit1 

If  tli,'  i^nit*  <n i',  I-!,',  ,t,nl  '•!•'  />''•  "n   ,i  xisitiiilit   Hi"    ".  f/"    .-•' 
lini'x  .1.1',  /;/;',  ,///,/  cc'  1,1,',-t  In  ,i  )><!/»(  o. 


n, I  cf1   //>'   i>n    it   at fii'ilit    //'//'    n. 


T\\0   DIMENSIONAL  (JKOMKTRY 


the    unit 


Take    -\  >   as   triangle   tit    reference   and    \      >•    as 

'"'"  1  f  /'^ 

1,0"     I"     Then  tin'  coordinates  of  j     |  arc  0:0:1,  those  of  |7)  | 

arc  0:1:  0.  those  of  ]  '  1  arc  1:0:  0,  and  those  of  \    '  \   are  1:1:1. 

f  I'l  r/n 

T>\     $  JS    the   coi'irdinates   ot   ]'     f   arc    1  :  1  :  1  -f  X,    those   of    <       > 

l<t  J  I/'  J 

arc  1  :  1  4-  /J.  :  1,  and  those  of  -j     ,  r  are  1  +  ^:1:1. 

The    coordinates    of    any     /  .  l")int   on   A'!'\     are    therefore 

Lline    through    f/'//J 

.....      f  point  lies  also  on  .1  />' 
1  +  p:  1  +p(  1  -f  /*):!  -f  X  +  /o,  and  if  this  1  l  ' 


,      , 
passes  also  through  ab 

f  re1  ~] 
we    must     have    p  =  —  1.      Hence    the    eoi'irdinatcs    of    <    ,  ,/  f    aro 

0  :  —  u  :  \.    Similarlv,  the  coordinates  of  S  ^   are  v  :  0  :  —  X   and 

•        ,  I  />'/,"/ 

-   J   ""    I 
the  coordinates  ot  1    <  «/  1   are  —  v  :  /*  :  0.    Since 


-/*          X 

0        -  X    =  0, 

u  0 


r  ^  fiit        t  ~\ 

I    points  a</     lili ,  a'    I  f 

trie  three  •(  ..  lt        ,  ,      ,    ,(  have  a  common   J. 

limes  j.C,  /;/;',  cc'j  \ 

two  theorems  are  therefore  proved. 

„,.       f  point  1  f   line  «   1 

1  he  <    ,.         /•  enuat  ion  ot  I  he    <        .  >    is 

t    line  J  I  point  r>J 


. 
point  () 


C  X/z./'j  -f-  i'\.r  ,  -f  pi'*.. 


-f- 


1 
J 


l-'or  the  complete  quadrilateral  we  shall  prove  the  following 
thci  ircm  : 

///.  A/IIJ  f'l'n  <l((i>fnnnlx  tit'  it  /•nniplt'ti'  nil  /til  riliitiTitl  hitfrx<:<it  tin1 
/////•>/  tfiuf/nmt!  in  tii'n  jHiinf*  //'///'•//  iiri-  lut  rinmi'ii'  rnnjtx/dft'K  f"  tin  ftrn 
i-i  rf/i'i-K  ii'fiu-fi  I  if  an  t/i/tf  (Hiii/miiil. 

In  I-'i^.  I:!  let  the  two  diagonals  /..Y  and  .I/A'  intersect  the  third 
diagonal  l'<t>  in  the  points  li  and  ,s'  respect  i\cl\'.  \\'c  are  to  prove 
that  A'  and  .s1  are  harmonic  conjiiLrates  to  /'  and  <t>. 

Since  liv  III,  ^  'Jit.  the  cross  ratio  is  independent  of  the  coordi- 
nate svstcm.  we  shall  take  the  triangle  IJ'<>  as  the  triaii'j-lc  of 


POINT    AND    LINK   COORDINATES    IN    A    PLANK         47 


reference  and   ihr   point    A'  as  tlic   unit   point,  so   that   the  coordi- 
nates  of    /'   are   0:0:1,    those   of    (t>   are    1:0:0,    those   of    /.    are 
0:1:0,    and    those    of    A'    arc     1:1:1.     Then    l>v    <  iM    it     is    easy 
to  sec   that    the  coordinates  o! 
I!  are   1:0:1,  those  of   .]/  arc 
0:1:1,  those  of  1\  are  1:1:0, 
and    finally    that     those    of    S 
arc     -1:0:1.     p,y    jj    11    the 

thcol'ein    follows. 

The  dualistic  theorem  to  III 
is  as  follows : 


(III  I'll     <1  til'JnlKll    l>nlltt,     tin'     (ll'n     )nl  III  //'/     /tHt'H     ill'l'     ]nl  I'll/nil  t ''     ft  i//  /  //'/'if  t'X 

fu    tin-    tii'n    x/'i/i-x    'it'  thi<    <niit<]  r«  in/li*    which   jxtxs    f/ir«n<//i    f/t>/f    tliir<l 
i]  nn/nii, il  point. 

The  proof  is  left  to  the  reader. 

Since  the  cross  ratio  of  anv  four  lines  of  a  pencil  is  equal  to 
the  cross  ratio  of  the  four  points  in  which  the  four  lines  cut  anv 
transversal  (^  In'),  theorem  I  \'  leads  at  once  to  the  following: 

V.  Tin'    xfi'iin/lit    Inn'    fu// tii'i'f  t /a/   ilili/    tim    il  nii/nini!    jmuitx    "t    it    t'ntn- 
jili-fi-   <[iini]  1'itin/li'  iiH-t'tx  tin'  s/'i/rx  at'  tin1  iiinnl  rii  iii/li1    /r/iii'/i    <!<>   n»t  JHIXX 

t  II  l'<illi/ll      til'1      til'"      i/ tili/n//il/    [mlHtx,      ill      /tl'H      lin/llfx      !/'llli'Jl      dl'i'      ll'l  I'/ll'iH  !>' 
rnnj {li/ilt,  X    1n     ill,'     tll'u    iliili/nii'll   jiniiltx. 

S'unilarlv,  theorem  111  ma\'  lie  ivplaceil  b\-  the  theorem,  dualistic 
to  \  .  as  tolli  iws  : 

VI.  //    tin'     Utti-rM'i-titill     'it'    >//!//     til'"     if/Hf/'ili'l/     Ihn'X    <>/    'I     t'ntitj,],t,' 

ijii'iil riliiti'i'iil    /x    i'»n ni'i'li -il    iritli    tin-    tiro    1't'i'firrx    <>}'   tin'    ijiiiiil r/l lit , •/•>// 

ll'llii'll      ll'i      lint      //','      nil      ///,        tll'n    lUililnllllla,      //I''      tll'n      l'<  i  II  1 1 1' < 'f  I  1 1 1 J      ///I''X      ll/'i 
//il  /'///«//  /(•     i-nllj  Ili/ilf  i  X     tn     tin'     tll'n     il'l, li]!, IK  I/X. 

Theorem  III  '_j'ivcs  a  method  of  tindin^  the  fourth  point  in  a 
harmonic  set  \\heii  three  points  arc  known.  In  I' IL;'.  1  •'>  let  us 
suppo.se  /',  O.  and  //  ^iven,  and  let  it  Ke  required  to  lind  >'.  The 
point  /.  may  lie  taken  at  pleasure  and  the  lines  /./'.  /./.'.  and  /.',> 
drawn.  Then  the  point  .V  mav  lie  taken  at  pleasure  on  /. //  and 


T\Yt>    m.MF.NSloNAL  GEOMETRY 


the  points  .!/  and  A"  determined  hv  drawing  (,>.V  and  7'JV.  The 
line  .I/A"  can  then  be  drawn,  determining  S. 

\\'e   \\ill    now   prove   the   following   theorem: 

VII.  Theorem  of  Pappus.  If  /',  /',  /'  <!/•>'  tlir«>  pnhitz  ,>n  a 
xffiii'////  fin''  iiii'l  I',  /,',  /,'  iifi1  tlif, -I-  pouitx  on  another  nfrttiy/it  l/>n\ 
tin'  tli,;,'  /'"iiitx  <>f  iiiff/'Ki'ffi"ii  <f  tin'  thri'i'  /><///•*  of  linrx  7,>/_'  and 
/4'/'.  /'/•  '""/  /'/,'•  I'Jl  '""I  I'Jl  h''  "»  «  ftrnii/Jit  /hi*'. 

\\'e  mav  so  choose  the  coordinate  svstein  that  the  line  contain- 
ing /.'.  /'.  /-'  (  I'"  i'_;'.  1-0  shall  be  .^  =  0  and  the  line  containing 
/.!.  /.'.  /,'  -hall  be  ./•,=  <>.  \\'c  may  then  take  the  line  /,'/_!  as  the 

line  ./•     =  'I,   so   t  hat    the  coi'irdi-  tj- 

/  \ 

nates  of  /,'  are  (H  :!:'))  and 
those  of  /.!  are  (  1  :  <•  :  H  ).  and 
mav  so  take  the  unit  point 
that  the  coordinates  of  /'.  are, 
(0:1:1)  and  those  of  /,'  ai'e  P, 
(  1  :  n  :  1  ).  (  'all  the  coiii'd  inates 
of  /'  ( il  :  1  :  X)  and  those  ol 
/,'  (  1  :  o  :  f_i  ).  Then  the  eijita-  — JT^T 
tii m  of  /;  /!  is  ./',  -  0  and  that 
of  /,'/'  is  ./•,  -f-  X./'.,—  ./'r-- ".  These 
lines  intersect  in  the  point 
A"  (X  :  -  1  :  (>).  The  conation 
of  /'  /'  is  ./-.,—  .r,  -  il  and  that 
of  /'/.'  is  /a./'  4- X./-o —.*•.,=  0.  These  lines  intei'scct  in  the  point 
L  (  \  -  \  :  fj.  :  /JL ).  1  he  ei jiiat  ion  ot  /'  /|  is  ./•  +  .''.,  —  .''.,-=  0  and  that  of 
/.'  /'  is  fj..' '  -  -  .r  --  ii.  These  lines  intersect  in  J/"  (1  :  /x  —  1  :  /z).  Since 

X  -1         'i 

IX  f.i  n.         I), 

| 

the  three  points   /..  A",   M  he  in  a  straight   line,  as  was  to  be  proved. 
Dnalistic  to  this  theorem  is  the  following: 

VHl.  If  /'..  />.,  /'_  <!,;•  t /I /•<'•'  Xt,-<li<lltt  HtH'X  tl/l-ntlitJl  'I  )><>;„(  flit'? 
l>  .  /'  .  i>f  ''/'•'  f fii'ii'  N/ /'////////  liin-K  ////•'///'///  tl/inf/it't'  jxii/it,  t/n'  f/i/'ii'  /tllrK 

'•"»»"•'<'><:/  'I-'   f !»•••''  /•'">•"  •;''  /""'/''•-'  /<,/',  <iH>1  /',/',-  }>.,/>,  <iH>l  /'./>,, 


POINT  AND    LINK  roulJDINATKS    IN    A    PLANK         4'.» 

EXERCISES 

1.  Prove  theorem   I  V. 

2.  Prove  theorem  \'1 1 1. 

3.  A  triangle  is  so  placed  that  its  vertices  /',  <j,  /,'  arc  on  the  sides 
AH,  A'',  and  !'><',  respect  ively,  of  a  fixed  triangle  and   its  sides  /'/,'  ;md 
A'','  puss  through  two  iixed  points  in  u  straight  line  with  .1.     Prove  that 
the  side  /'(,*  pusses  through  a  iixed  point. 

4.  A  triangle  is  so  placed  that  its  sides  (>/',  /'/,',  r<>  pass  through 

the   Vertices    (.',   />',   .1,   respect  i  Vel  v,  of   U   1'lXed    triangle   and    its   vel'tiees    'I 

and   /'  lit1  on  two  iixed  lines  which  intersect  on    1><  '.     Prove  that  tin- 
vertex  A'  lies  on  a  straight  line. 

5.  Given  a  straight  line  y/  and  two  iixed  points  .1  and  I',.    Take  any 
two  points  on  p  and  eonneet  each  of   them  with  .1   and  //.    These  lines 
determine  two  new  points  ('  and  1>  bv  their  intersections.    Prove  that 
the  line  <'!>  pusses  through  a  Iixed  point  on  A/!. 

6.  (liven  a  point  /'and  t  wo  Iixed  lines  u  and  f>.    Draw  anv  two  lines 
through  /'  and  connect  their  points  of  intersection  with  n  and  />.    This 
determines  two  new  lines  <•  and  '/.    Prove  that  the  point  of  intersection 
of  i"  and  <l  lies  on  a  Iixed  straight  line  through  "?,. 

7.  Three  lines/',  y,  li  are  drawn  through  the  vertex  .!  of  t  he  triangle 
All''.    On  (i  any  point  is  taken  and  the  lines  /and   ///   are  drawn  to  f 
and  Jl  respectively.    The  line  /  intersects  /'  in  1>  and  the  line  ///   inter- 
sects }t  in  I'..    Prove  that  1>1'.  passes  through  a  Iixed  point  on  Hi'. 

8.  Three  points  /'.   <i,  II  are  taken  on  the  side   J}< '  of  the  triangle 
AI><'.    Through   <i  any  line  is  drawn  cutting  All  and  AC  in    /.  and  .17 
respectively.     The   lines   /'/.   and    JIM   intersect    in    A'.     Prove   that    the 
locus  of  A"  is  a  straight  line  through  .1. 

9.  Show  that  if  n,  u'  and  A,  //'  are  any  two  pairs  of  corresponding 
lines  of  two  protective  pencils   not    in   perspective,  the   line  connecting 
the   points  nli'  and  "'I*  passes  through  a   tixed  point.     This  is  called  the 
<-!  n  fir  >if'  ]/ni,ii>ln,/ii  of  t  he  two  pencils.    Showtliut   it   is  the  intersection 
of  t  he  two  lines  which  correspond  to  the  line  connecting  tin1  vert  ices  ot 
the  pencils,  considered  as  belonging  tir-i  to  one  pencil  and  then  to  tip- 
other. 

10.    Show  that    if  .  I.  .1  '  and   /,'.  //'  are  anv  two   points   of  two  pn.j 
tive  raiiL,re-'   which  are   not    in    perspective,  the   point    of  intersection   of 
the   lines  All'  and    .!'/>'  lies  on  ;i   fixed  straight    line.     This   is  called   the 

a.ri.<  nf  Jmiiinln'ii/  of  t  lie   t  wo   railLfes.      Show  that    it    intersects    the    I'Usenf 

each  range  in  the  point   which   corresponds  to  the  point  of  intersection 
of  the  two  liases,  considered  as  belonging  to  the  other  range. 


f,|)  TWO    DI.MKNSIONAL  (IKO.MKTKV 

31.  Curves  in  point  coordinates.    The  equations 

J\:  s.,:  .r,  =  <£,(/)  :  <M  '  >  :  <J>,(0,  (V) 

\\here  /  is  an   independent  variable  and  the  ratios  of  the  functions 


arc  not  constant  or  indeterminate,  define  a  one-dimensional 
extent  of  points  called  a  citri'i'.  It  is  not  necessary  that  any  point 
of  the  cur\e  >hould  lie  real.  We  shall  limit  ourselves  to  those 
curves  for  which  the  functions  <£,(/)  are  continuous  and  have 
derivatives  of  at  least  the  lirst  order. 

If  4>,(t)   is  identically  /.ero  the  curve   is  the  straight  line  -''3=  0. 
Otherwise  we  may  write  equations  (1  )  in  the  form 


It   is  then   possible  to  eliminate  t  between  the  equations  (-)  with 

the  result. 

11  =  cD    •' 

V* 

Conversely,  let   there  be  given  an  equation 

where/' is  a  homogeneous  function  in  ./•..>•„.  r..    I»v  a  homogeneous 

function  we  mean  one  which  satisfies  the  condition 

/'(X./-J.  X.r,.  \sa)—\"f(.ri,  ./;,,  r;!). 
where  X  is  any  multiplier,  not  zero  or  infinity.    In  particular,  if  we 

place  X    -        we   have 
./• 


lor  all  points  for  which  r    is  not  /.em.     Kquation  (4)  mav  then  be 
writt<i|1 


h 


\\  e  shall  limit  ourselves  to  functions  /'\\'luch  are  continuous  and 
have  partial  derivatives  of  at   least  the  lirst  order. 

\\'e  shall  also  ;is>unie  that   (  {  )  is  satisfied   bv  at   least   one  point 


POINT  AND   LINE  COORDINATES  IN  A  PLANE        ;",  1 

does  not  vanish.     Then  similar  conditions  hold  for  (•'>),  and  by  the 
theorv  of  implicit  functions  '   we  have,  irom  (•>), 

,S'    =3 


which   is  valid  in  the  vicinitv  of  t  =  —  ,   *  =  —  '• 

//a        °       ^, 

1  his  last  equation   mav  he  written 


which  is  of  the  type  of  equations  (  1  ).  Hence,  under  our  hypotheses 
('([nation  (4)  represents  a  curve. 

The  above  discussion  leaves  unconsidered  the  points  for  which 
j-  =  0.  These  mav  be  found  bv  direct  substitution  in  (4  )  or  we  mav 
repeat  the  discussion,  dividing  by  some  other  coordinate,  perhaps  j^. 

Let  /'(//  : //., ://..)  be  a  point  of  (1)  corresponding  to  the  value 
/  =  f  .  and  let  (,M//+A//  :  //, +  A// .,://..  +  A//., )  be  a  point  correspond- 
ing to  t  +  A/.  These  two  points  lix  a  straight  line  with  the  equation 

the  coefficients  of  which  are  determined  bv  the  two  equations 

ttl  ( >/1  +  A //, )  -f-  «.,  ( //.,  +  A//., )  -f  it..  ( i/.,  +  A// . )  ==  0. 
From  these  it  follows  that 

It  is  to  be  noticed  that  these  involve  the  ratios  of  the  in- 
crements A//,  A//,.  A//..  If  now  A/  approaches  /.ero,  ihe  point 
(,>  approaches  /'.  the  ratios  A//^  :  A//., :  A//,  approach  the  ratios 
i///j  :  <lif^ :  it  if  ,  and  the  ratios  ft  :  <ta:  it.,  approaeli  the  limiting  ratios 


:  n  .= 


The  straight  line  (  ti  )  with  the  cocl'iiciciits  delined  bv  (  >s  )  is 
the  limit  of  the  secant  I'*,)  and  is  called  the  t<i/i</>-nt  to  the  curve. 

If  the  ('([nation  of  the  curve  is  in  the  form  (4),  the  equation  (if 
the  tangent  mav  be  modified  as  follows: 

Since    .'(//,://.,://.,)    i.>    a    homogeneous    function    we     have,     bv 

Kll  lei'  S    1  heorelll, 


2  TWO-DIMENSIONAL  (JKOMETRY 

()n   the  other  hand.  '///^  <///.,,  <///.,  satisfy  the  condition 

iJf=   ;/-^//1 4- ';;<///.,+ ^^3=0.  (10) 


the  ec  1 1  uit  ion   nt'  the  tangent   line   is,  from  (S)  and  (Id), 

rf'  <•  /'  of 

.'•.    '    +  .'•.,   '    +  J\.    '    -     «>.  (II 


=  0,  =0,  =0.  (12) 

Points  for  \\  hirh  the  conditions  (1  2)  hold  are  culled  xin<inl<ir  /mint*. 
\\e  mav  --11111  up  as  follows  :  At  rt'eri/ nutusiiit/ultii' jmtHt  (>/  :_//.,://.) 

/'(./'.  ./'  ,  ./'    )  =  (') 

•     v      1         -J         ;t  ' 

///,/v  />•  ,/  :/,/initt'  tititi/i'ttt  l//n'  i/ti'i'/i  /»_//  ///<•  w [nation 

(f  (f  cf 

('onsiiler  now  anv  straight  line  dt'terinined  hv  two  fixed  points 
//  and  ~it  so  thai  if.-\-\zi  is  any  point  of  the  line.  The  point  //t  -f  \z 
lies  on  the  cum-  (  1  )  when  X  has  a  value  satisfying  the  equation 

\\hii-h  expands  hv  Tuvlor's  theorem   into 

-I+JA  +  J.X-  f-  ...  =  <»,  (14) 


It  //   is  on  the  eii  r\  c  (  !  ),  .1      :  (I  and  one  root  of  (14)  is  /.en>.    If, 

in  addition,   .1        "  and   //    is  not  a  singular  point,  ^.  lies  on  the  tan- 

^'•n i    line  to  (  1  )  and  two  roots  ol   (1  1  )  are  /ero.     It   //,  is  a  singular 

'    of   the   curve,     I        "  and    ./=-()   for  all   values  of  ,r  :    that    is, 

•    i:ii    it  >•///. /I//*//-  i><>ii,t  ><!'•/  <•//,•'•,    ittti'/'xi-clx  (/»•  i-nri'i'  /n  <(( 

•     •  ,•  ,///,-/-/,  ,,(  [...'nits. 


POINT   AND    LINK   C'OOlUHN  ATKS    IN    A    PLANK         '>'.} 

If  t\.r  ,  ./•„,  .r.  )  is  a  homogeneous  polynomial  ot  the  //th  degree, 
the  locus  of  points  satisfying  (4)  is  defined  us  a  ,•/>/•/•>•  t,t'  ///>•  nth 
oril<r.  Equation  (14)  is  then  an  algebraic  equation  of  the  //th 
degree  unless  its  left-hand  member  vanishes  identically  for  all 
values  ol  A..  Hence  dtiif  ruri't'  »f  tin1  nth  »/••/>'/•  t*  <•///  /•_//  <mt/  xtr<<i<//tt 
Itni'  t/i  n  point*  mili'xx  tin'  itf/'<n'i//i(  Inh'  lift  1'iit  iri'l  ;/  <>/i  t/n'  fiii'i.'f. 

32.  Curves  in  line  coordinates.    '1'he  eiiiations 


\\here  t  is  an  independent  variable  and  the  ratios  of  the  functions 
(/>,(/)  are  not  constant  or  indeterminate,  define  a  one-dimensional 
extent  of  straight  lines.  We  shall  see  that  these  lines  determine 
a  curve  in  the  sense  of  sj  :>1.  Equations  (1  )  are  called  the  line 
equations  of  that  curve. 

Proceeding  as  in  ^  ol  with  the  same  hypotheses  as  to  the  nature 
of  the  functions  </>l(^)'  WL-  lliav  sho\v  that  equations  (1)  are 
equivulfiit  to  the  e(]iiation 


('(jnverselv,  let  there  be  given  an  equation 

/(  >/i?  //,,  //  ,  )=  0,  ('!) 

where  f  is  a  homogeneous  luiiction  in  n  ,  it,,,  u  .  :  \\'e  mav  show,  as 
in  vj  -Jl.  that  c»jiiation  ('!)  deiines  a  one-dimensional  extent  of  lines 
of  the  type  (  1  ). 

The  discussion  now  proceeds  dualistically  to  that  in  ^  -\\. 

Let  y(  i'  :  '•„:  r..  )  and  y  (  r^  -f-  A/^:  '•„  f-  A'\,:  '',  —  A/1.,  )  be  two  straight 
lines  determined  bv  placing  /  /  and  /  /^+A/  in  (1  ).  Thoe 
two  lines  determine  a  point  l\  the  coordinates  of  \\hich  >ati>fv  the 
t\vo  eijuations 


(  /•  +  A/'  ').<•+(  '•„+  A/-.. 


Now  let  A/  approach  /ero.    The  line  >/  approaches  ihe  line  /-.  ih 
ratios  A'1  :  A/1., :  A/1.,  ajiproach  i  he  rat  ios  !//•;,/,•;,/,•.  ain  i   t  he  pi  >m 

K  approaches  the  point    L.  <>t   \vhich  ihe  coi'irdinates  are 
ii  i 

.-    :  j-:  ./•-     >•,/,•    -  r  ,/r  :  <•  ,/,•  ,  :', 


~)4  T\V()-J)LMKNSI()NAL  (JKOMKTKV 

1>\  \irtue  of  (J> )  and  (  1  )  the  points  /,  form  in  general  a  curve. 
An  exception  would  occur  \\hen  the  I'ight-hand  ratios  of  (^> )  are 
independent  of  /.  In  that  case  the  points  L  for  all  lines  of  (^1) 
ei  uncide. 

If  the  extent  of  lines  is  defined  by  a  single  equation  (-!)  the 
coordinates  ot  /,  mav  be  put  in  another  ionn,  as  follows:  Sillee/ 
is  a  homogeneous  function  we  have,  hv  Killer's  theorem. 

(f  <  /'  <  /' 

'     '',+   '     ''.,+   '     i'    =  >(t  =  0. 
(  /•  t  r^    '      (  t\ 

<  f  (  f  f  f 


'1  In-  coordinates  ot    /.  are  theretore 

ft'     <  f     ( 


(  h 

These  equations  determine  a  tiniqiie  point   on  anv  line  j>  unless 


borlio-id  of  >\.    This  would  happen,  for  example,  if 
/  —  ( '',",+  ".,".,  +  "..",  )</>(  ",.  ".,,  ".) 

and  i\  is  anv  point  which  makes  the  first   tactor  vanish.    The  points 
/.  on   all   lines   in   the  neighborhood  of  /•.  are  then  all   <(  :  <t±i:  if... 
Leaving  the  exceptional   case  aside   we  have  the   theorem: 

f  hi  iiiii/   ii'inxiiii  /uliir  I/HI'  "/'  </   niii'-<l  i  an' iixiniiitl  i  '.rti  nt   nt    Inn-*  tin  ri' 

Inxil  iliil'iiii'  i>"iiit,  i-ill/ii/  il  ]  1 111  il  j>"l  lit ,  tin1  /HI-US  nt  (I'll  It'll  IX  til  i/r/li  /'i// 
>l  t'lli'i'i;  77//.V  rU/'l't'  ix  xiliil  tu  /„•  1/1  /i/ni/  in  I'm,'  <-unr<i  i  lull  i  x  In/  tin' 
i  ijiiiit ii.n  nt  tin-  I/in'  i.rtint.  In  xini-iiil  fiixi-x  tin'  r/iri'i'  unit/  rid iii-i'  In 
'i  I'-i/nt  a/-  i-., ///,////  </  iitiiiilnr  ni  ji'iinlx  tin  jHirtx  nt  tin'  i-ii/'i'i1. 


I'nlNT   AND    LINK   OxiRDIXATKS    IX    A    1M.AXK         -V". 

In  case  we  have  a  true  curve  of  limit  points  it  will  he  pos>ihle 
to  solve  equations  (  I;  for  r  :  /•_:  r  and  sulistitute  in  (-).  This 
'_d\es  .  , 

\\liich  is  the  equation  in  point  coordinates  of  the  locus  of  A. 

From   ( .) ), 

(  r       ii 
i 

'-,"'  +•>:," -  +  •>•.,' 

i  .r       ~  i  ./•       "  '  ./• 


~  P'\- 
i ./'. 

This  shows  that  the  tangent  line  to  the  curve  (.•>')  at  the  point 
/.  is  the  line  /'.  Ileiiee  \\'e  ha\c  the  theorciii: 

J'Jili'/l  lilli-  <>t  il  li/(t'-<It'ltU'lt#tfilHtl  I. l'l>  III  »_t  I/ltt'X  IX  litlUjiItt  (it  tt* 
Hunt  iii-nit  t"  tin'  citt'l'i'  H'li/i'Ji  IX  tin1  IIII-IIH  nt  tin'  limit  [><>intx.  Tin' 
liin.-:  ///-/•«•/'.-/•,•  ftti'ffuj)  t/li'  i-Ui'i'i'. 

Li'i  us  suppose  now  that  in  equation  (  -  )  t/"  is  an  algebraic  jiol v- 
noinial  of  the  //ih  decree.  Then  the  locus  nf  the  limit  points  /.  is 
called  a  i-it/'i'f  nf  tin'  /if//  f/itx*.  \\'e  shall  pi'o\ c  that  ////•"//;///  <<n</ 

jin'lllt     "t'    ///c    jililili'     i/u     II     /i/K'X     till/'/i'/lf     f"     'I     rll/'l'l'     "f    tin'     lltll     i-tilsK. 

lo  do  this  \ve  have  to  sho\\~  that  //  lines  sat  1st vm^  e(|iiation 
('!)  '4'o  throii^'li  anv  point  o|  the  plane.  No\\-  anv  point  is  li\cd 
h\  t  \\  o  lines  '•  and  tr.,  and  an\  line  tlil'tiii^Ii  that  point  has  the 
eoi'irdinates  •;  -f-  \>/\.  This  line  satisfies  (  i' )  \vhen  \  satislies  the 
•  •I I uat  ion 


This  is  an  equation  of  the  //ih  decree,  and  the  theorem  is  proved. 

\\'e  ha\c  shown  in  this  section  that  a  one-dimensional  extent 
of  lines  are  in  general  the  tangent  line-  to  a  curve.  ( 'on verselv. 
the  tangent  lines  to  an\'  curve  are  rasilv  sliown  to  he  a  olie- 
diiuensioiia]  extent  of  lines.  An  exception  occurs  onlv  \\heii  the 
curve  consists  of  a  nnml»T  o|  strai'dil  lines. 


TWO-DIM KNSIl )XA L  (IE( >M ETKV 


The  dualistic  r^Uuion  between  point  and  line  coordinates  is 
exhibited  in  the  following  restatement,  in  parallel  columns,  of  the 
results  nl'  ^  ill  and  ol' : 


An  ec[iiat  ion  /(.'',.  •''.,.  .''.j)  :=  ^  is 
xiti-died  hv  a  one-dimensional  ex- 
tent of  points  which  he  on  a  curve. 
A  line  jiiining  two  consecutive 
points  cit'  the  curve  is  tangent 
to  the  curve.  Its  line  coordinates 

cf     cf     cf 

are   ii   •.  ii  ,:  a,    -     '  -  '•         '•  .       •     ine 

C~'\       C^2      °'3 

elimination  of  ./•:./•.,:  r  between 
these  equations  and  that  of  the 
curve  gives  the  line  equation  of 
the  curve. 

The  equation  of  the  tangent 
line  to  the  curve  defined  by  the 
1'nint  extent  is 


An  equation  ^/"(MJ,  ».,,  »8)  =  0  is 
satislied  by  a  one-dimensional  ex- 
tent of  lines  which  are  tangent  to 
a  curve.  A  point  of  intersection 
of  two  consecutive  lines  is  a  point 
on  the  curve.  Its  point  coordinates 

£/  .  cf     cf 
jc,:  xa=  ~  •  -  --     1  he 


elimination  of  «]:».,:i/3  between 
these  equations  and  that  of  the  line 
extent  gives  the  point  equation  of 
the  curve. 

The  equation  of  a  point  on  the 
curve  enveloped  bv  the  line  ex- 
tent is 


If./1  is   of   the   ?/th   degree   the 
curve  i>  uf  the  n\  h  order. 
<  )n  an     line  lie 


If  f  is   of  the    »th   degree  the 
curve  is  of  the  ?<th  class. 

points  of  the  Through  any   point   go   n  lines 

which  are  tangent  to  the  curve. 

The  curve  of  the  first  class  is 
a  straight  line,  the  base  of  a  pencil  a  point,  the  vertex  of  a  pencil  of 
of  points.  It  is  of  zero  class  and  lines.  It  is  of  zero  order  and  has 
has  no  line  equation.  no  point  equation. 


curve. 

The  curve   of  the   iirst  order  is 


EXERCISES 

'ind  the  >ingular  point  of  jrf  -f  .rf./-3  —  .r.;./-(  =  0.  Show  that 
a  the  singular  point  go  two  real  lines  which  meet  the  curve  in 
oincidi-nt  points.  Sketch  the  curve  with  special  reference  to  its 
i  with  the  t  r;;tii'_;it"  of  reference.  Also  sketch  the  curve  interpret- 
•  coordinates  as  ('artesian  coordinates  and  taking  jr.  =  0,  -''._,—  0, 
successively  as  the  line  at  infinity. 

'mil  the  >iii!_Milar  point  of  J-,H  —  j-,r>.,  =  0.     Show  that   through  it 
oincident  lines  which  meet  the  curve  in  three  coincident  points. 


POINT  AM)   LINE  COORDINATES   IN  A  PLANE         07 

3.  Kind  the  singular  point  of  the  curve  ./-j!  -f-  rf.i-.j  -f-  .rrr.j  =  0.    Show 
that  through  it  go  two  imaginary  lines  which  meet  the  curve  in  three 
coincident  points.    Sketch  the  curve  as  in  Ex.  1. 

4.  Kind  the  line  equation  of  each  of  the  curves  in   Exs    1    -'?. 

5.  Show  that  anv  point  whose  coordinates  sat  isfv  the  three  equations 
=  0,  =  0,  —  0    lies    on    the   curve    /'  —  0   and    is   therefore   a 


Cartesian  coordinates  are  unven   hv    '     —  0,    '  -  =  0.  provided  the  solu- 

-      CJC  Ci 


singular  point. 

G.    Show   that   the    singular    points    of    a   curve    in    nonhomogeiieous 

'  '  - 

iJ 

tions  of  these  e(|uatioiis  also  satisfy  f(j",t/)=Q.  (Compare  Ex.  5.) 
Applv  to  lind  the  singular  points  of  .<•-  -f-  //"  =  «'  and  ./••  —  //~  =  (). 

7.  Show  that  through  anv  point   mi  a  singular  line  of  a  line  extent 
go  at  least  two  coincident   lines  of  the  extent.     Hence  show  that  if  the 
extent    envelops   a   curve   of   the    n\\i    class,    the   singular   lines   are   the 
locus  of  a  point  such  that  at   least  two  of  the  n  tangents  to  the  curve 
from  that  point  are  coincident.     Illustrate  liy  considering  the  line  extent 

//•!  -f-    H.,lf.~  =   0. 

8.  Il'y'i./1  ,  ./•.„  ./•„  )  =  0  is  the  equation   of  a  curve  and   //  :  //.,  .  //.    is  a 
lixed  point,  show  that  the  equation 

ct'  (  f  cf 

//,      -  +  .'/.,  r-  +  ;/.,    '     =  0 

l(''\  '^o  ^'•'•, 

I'ejiresents  a  curve  which  pas>cs  through  all  the  singular  points  of 
f  —  0  and  through  all  the  points  of  taiinvncv  from  ,/t  to  /'--  0.  lait 
intersects/'—  0  in  no  other  points. 

9.  Prove  that  a  curve  of  the  third  order  can  have  at  most  one  singu- 
lar  point   unless   it    consists   of  a    straight   line   and   a   curve  of   second 
order,  or  entirely  of  straight  lines. 


\ 


fTKYKS  <»F  SK<  <>\n  ni;l>Kl;   AND   >K<  <>\j)  ''LASS      .Y.< 

point  0  :  "  :  1 .  Th»-  d'-'jT'-*-  nf  th.-  t-ipiatinn  will  imt  U-  > -l.ai^-'i 
fi  L".(  >.  Imt  in  th»-  nt-'.v  »-ijiiati"M  w>-  -hail  hav»-  </  =  ",  ./_  =  <.». 
./  —'i.  'I'liL-  t-.jnatinn  tht-n-fnpt-  U-fi,m»-> 

whi'-h  '-an  }>«•  factored  intn  t'.\'<>  lin>-ar  fa<-t"!--.  'I  i;>---  f,i--t<ir>  >'an- 
imt  }»•  .-(jual.  fnp  if  tht-v  U'-r--  we  >li-iiild  hav»-  -/.  :  •/  :=  -/  :  ./  .  a:-i 

\\-"u!d  liavt-  iimr.-  thaii  (.rn-  -njiitinn.  H.-n'-»-  th».-  1m  u-  "f  <  1  ,  .,,;,. 
sists  nf  two  inteix-'-tiny;  -ipa:'_r:.t  lin--<. 

(  'A>K  III.  />  =  '».  iiri'l  all  its  ii:>t  ininnj->  a:>-  x^rn.  A:.v  sfluti-.tn  nf 
on'-  nf  tin.-  fjnatinn-  r  '1  >  i-  a  snlutinn  of  th--  ntln-rs.  an-i  th'-  '-urve 
ha-  a  lint.-  (.if  -insular  pn-nt-.  If  },y  a  f-hanurf'  "f  'if-«"'rdinate-  that 
lint-  i-  takt-n  a-  tli»-  lint-  r  =  0,  w.-  -hall  havt-  in  th>.-  n>-'.v  eijua'inn 
,in=  n  =  ,i  —-'i  — '/  =  0.  aii(l  tilt/ (-ijuation  tn-i-nines /.•='.'.  !{•:.'- 
in  thi-  i-a-t.-  ih--  curv»,-  ronsi-t-  nf  twn  coincident  straight  line-. 

Suminincr  up.  w»-  hav»-  th--  fnllnwincr  theorem: 


///-//  j,',u,t.     If  if   /,,/.y   ,/    //,,.-   -,f   ginijul'.ir  j« 

,/„,//,/,/  r  >•<•!;'.  n«L 

Tlit-  fiirvfs  '  if  st-fonil  <ir'l»-r  in  lionii'^ii^nus  rr>i'iril!natv>  ar--  if.'- 
saint-  a»  til-'  coni'-s  in  ('artr>;an  ci  H"irilinat>-<.  fur.  as  slid\vn  :;i  ;j  i!'.'. 
tin-  il»"_frfi-  i  if  an  fijuati'"'n  'N  nut  alti-ri-'l  bv  a  rhali'j*-  <if  r»<">r<I::.atf-. 
\V'-  niav  on  ocrasi'in  distinguish  lictwf.-ii  th.-  cnnii-s  \vith"iu  >::._"!- 
lar  ji'iints  and  thnst-  \vhic)i  musist  "f  twn  s;rai'_rlit  lint-s  ov  ca'.linu' 


34.  Poles  and  polars  with  respect  to  a  curve  of  second  order. 
\  Mil.  i  '•'>],  if  //,  i>  a  piiint  "ii  ili».'  cniiir  (  1  ).  s-  '.]'.],  tilt-  lint.- 
i'ii-ilinates  of  tlui  tan^'-nt  at  //  ar>> 


00  TWO   DIMENSION  A  I,  (JKOMKTKY 

Equations  (  1  )  then  associate  to  any  point  //,  a  definite  line  »,. 
This  line  is  called  the  /<"/<'/•  of  the  point,  and  the  point  is  called 
the  pole  of  the  line.  The  equation  of  the  polar  is 


If  //,  is  given.  ?/,  is  uniquely  determined  l>v  (  1  ):  hut  if  xt  is  given, 
//,  is  determined  onlv  when  equations  (1)  can  he  solved,  that  is, 
when  the  discriminant  I>,  ^  •>:>,  does  not  vanish.  Hence, 

I.  T"  mil/  jmfttf  i>t'  fin'  plane  cnrrt'itp'inilx  ctUcni/s  »  unique  poliir  ; 
hut  to  any  litif  i  if  t)n[  jilttm'  fitrmtpomli*  >t  uni<pi<'  jm/,-  ir/nn  <tn<l  >.>nhf 
u'hsn  thi'  I'urri'  hux  n<>  siw/uhtr  point. 

The  following  theorems  are  now  easilv  proved: 

//.  Tin-  pnJiir  ii  f  <t  i>i»'»t  <>n  the  cnri'f  it  f/n'  f<oi</r)if  li>n>  cif  tJi,it 
point  <tn<l*  t'tinvi'wli/,  f/n'  p»h'  <>f  nn>/  t<iti;/i'»t  t»  tin-  rur>'i'  /x  t/n'  point 
of  1'ontni't  of  tl,t'  1nn;/ent. 

It  is  ohvious  that  equation  ('!)  reduces  to  the  equation  of 
the  tangent  when  the  point  //,  is  on  the  curve,  ('onverselv.  if 
equation  d)  is  that  of  a  tangent  to  the  curve,  the  solution 
of  equations  (  1  )  will  give  the  point  of  contact. 


///.   Tli'  politr  of  a  pnhit  pnxHf*  tJn'oui/Ji  //>,•  p 

irjn-l)    tin    pniiit    IK  o/l    f/n'   I'lirri'. 

This  follows  from   the   fact    that    the   substitution  .'',—  //,  reduer's 
equation    (  '2  )  to  the  e<|uation   of  the  curve. 

IV.      Tin'     /"'A//'      nf    II  II  >f    poillt     pitKKlK     f///-o/l<///      till'     Xl'/l'/lt/it/-     jiu/ltfx     /if 

tin-  <>>/r>'i    tt   xiirJi  i  j'ittf. 


V.   If  it  fi'iinf   !'  Ii,  x  "//   tin-  jxiliir  if  'i  point    <t>.   tin  n    (J  //,-*  ,,n   th? 
lnr  o     /'. 


CURVES  OF  SECOND  ORDER  AND  SECOND  CLASS      G 
If  /'is  the  point  y.  and  (t)  is  the  point  z, ,  the  polar  of  /'  is 

and  that  of  (J  is 

Tlie  condition  that  /'  should  lie  on  the  polar  of  <t>  is 


which  is  just  the  condition  that   <t>  should  lie  on  the  polar  of  /'. 

VI.  If  '7  curve  of  seennd  ori/er  )mx  tin  mniiulcir  i>oinf,  fn'»  t/ini/,  ///x 
.  »  • r          i  • ' 

niai/  he  eini  it'll  to  the  cur>'e  fr»m  iitii/  point  not  n>i  if,  ,/,/,/  f//,'  i-h>.,;/  ,•,,//. 
net'tini/  the  points  of  cont/ict  of  them'  t<ni//enfx  /x  the  pohi  ,•  ,,f  fh,-  point 
of  intersection  of  the  tan</entn. 

Let  P  (Fig-  !•">)  be  a  point  not  on  the  curve.  The  polar  of  /', 
being  a  straight  line,  cuts  the  curve  in  two  points  T  and  N.  These 
two  points  are  distinct  because  by  theorem  II  the  polar  is  not 
tangent,  since  P,  by  hypothesis,  is  not 
on  the  curve. 

Since  by  hypothesis  the  curve  has 
no  singular  point,  it  lias  a  unique 
tangent  line  at  each  of  the  points  T 
and  ,V.  These  tangents  are  the  polars 
of  their  points  of  contact  and  hence  by 
theorem  Y  pass  through  /•*.  The  polar 
of  P  therefore  passes  through  T  and  ,S' 
(theorem  Y). 

There  can  be  no  more  tangents 
from  P  to  the  curve,  for  if  there  were, 

the  point  of  tangency  would  lie  on  7'.S'  by  theorem  Y,  and  hence 
TS  would  intersect  the  curve  in  more  than  two  points,  which  is 
impossible.  The  possibility  that  TS  should  lie  entirely  on  the  curve 
is  ruled  out  bv  the  fact  that  in  that  cast1  the  curve  would  consist 
of  two  straight  lines  and  would  have  a  singular  point,  which  is 
contrary  to  hypothesis. 

This  theorem  as  proved  takes  no  account  of  the  reality  ot  the 
lines  and  points  concerned.  In  the  case  in  which  it  is  possible  to 
draw  real  tangents  from  /'.however,  the  theorem  furnishes  an  easy 
method  of  sketching  the  polar  of  /'. 


TWO-DIM  KNSIONA1.   (JKOMKTKY 


When  real  tangents  cannot  be  drawn  from  /',  as  in   Fig.  1*1,  the 
po'ar  of  /'  mav  be  constructed  as  follows: 

Through  /'  draw  two  chords,  one  intersecting  the  curve  in  the 
points  1!  and  N  and  the  other  intersecting  the  curve  in  the  points 
'/'  and  I'.  Draw  the  tangents 
at  the  points  /.'.  N.  '/',  and  /", 
and  let  the  tangents  at  //  and  S 
intersect  at  /.  and  let  the  tan- 
gents at  '/'  and  /'  intersect  at  A". 
Then,  hv  theorem  VI,  /,  is  the 
pole  of  A'\  and  A"  is  the  pole 
of  '/'/'.  Consequently  the  polar  of 
/'  passes  through  L  and  A'  and 
is  the  line  Ll\. 

VII.  For  a  i-urre  of  wvml  order 

iriili'iiit  xiii'julitr  jiointx  it  in  possible 

in  iin  intiniti'  tnunbi'r  <>t   tfiii/x  f/>  cotixtnti't  truni'ilt'n  in  ii'h/i'Jt  i  </>•//  ,sV'/»' 

/X    f//i'    Imliir  o1    tin'   njijxiXltt'    I'l'/'fl'J'.       Till  Hi'   il/'i'    I'llllnl  !<i  1 1 -jinjil  t'  ( t'/'l  >l'//i'X. 

We  may  take  A  (Fig.  17),  any  point  not  on  the  curve,  and 
construct  its  polar,  which  will  not  pass  through  -I  (theorem  III) 
and  cannot  lie  entirely  on  the  curve, 
since  the  curve  has  no  singular  point. 
We  may  then  take  /•'.  any  point  on 
the  polar  of  -I  but  not  on  the  curve, 
and  construct  its  polar.  This  polar 
will  pass  through  .1  (theorem  V  )  bin 
not  through  //  (theorem  III).  The 
two  polars  now  found  are  distinct 
lines  (theorem  I>  and  will  intersect 
in  a  point  <'.  Draw  All.  Then  All  is 
t  he  polar  of  f  hv  t  heoivni  V.  The 
triangle  .l/:c  is  a  self-polar  triangle.  I-'i-..  l~ 

VIII.  If    'i, a/    xtfui./Jit     Inn      ///     /x    IKIXK,'<I    fJif"ii,//i    ,i    /».////     / '.     itinl 

I!  ilinl  .S'  lift'  tin-  jin'nitK  nt'  illff't'Xt'ff/on  of  ,n  iritli  ,1  i-Ht'l'i'  nt'  tin' 
Xn'nllll  "I-'I,  l\  Itinl  <f>  /N  tin-  jin'nit  <lf  il/f,'rXl'l-f/"ll  <f  III  ll'itll  ill,  /.'[.If 
of  I'.  tk''tl  /'  il/nl  O  «//•/•  //-//•///"///<•  I'liHJIIi/tlti-K  ll'itll  fi'X/it'ff  fa  /i 

ami    N. 


(TKVKS   OF   SKCOND   ()i;i>KK    AND    SKm.ND  CLASS      (\l} 

Let  /'  (  Fig.  1  s  )  be  any  point  with  coordinates  //,,  let  />  be  the  j»i >lar 
df  /',  and  let  m  In-  any  line  through  /'  cutting  //  in  (tt  and  the  curve 
in  /<'  and  .s'.  'I  lien,  it  ~.  are  the  coordinate s  ot  <t>,  the  coordinates  of 

//  and  .s'  are  //i  +  \  .?.  and  //  +Xpr.  where  \t  and  X,  are,  the  runts  of 

the  equal  ion 

- 


ibtained  by  substituting  j'.=  >/.-+•  \*.  in  the  e(|uation  of  the  curve. 


Flu.  is 


I>ut    since    o   is   on    tin1   polar   of   /',  wi1   liave    ^  .",-<//,  ^  =  "•>   ;ll|d 

tlieri'foru  X       —  A..,.    I>\'  vj  1  t  the  thforcin  is  proved. 

'I  his  theorem  j^ivcs  a  method  of  liiiflilig  the  [tolar  of  /'  when 
the  curve  of  second  order  consists  of  two  st might  lines  intersecting 
in  a  point  f>  ( Fig.  li* ).  Draw  through  /*  tiny  straight  line  ///  inter- 
secting the  curve  in  the  [mints  J>  and  X,  distinct  from  < ),  and  tind 
the  jioint  ',',  the  harmonic  conjugate  of  /'  \\ith  respect  to  /,'  and  \. 
l>v  theoi'eiu  \'III,  it>  is  on  the  j)olar  of  /',  and  bv  theorem  1\  tlie 
[tolar  of  /'  passes  through  < >.  Hence  <)  and  <>  determine  the  i'e- 
([uired  jmlar  />. 

EXERCISES 


accordiiiLT  to  tlie  nature  of  the  eoellieieiils  rr 

'2.    I'ruve  that    it'  the   triangle  ol'   rei'i-rein-e    is  composed   of  two  tan- 
gents to  a  conic  and   the  ehonl  of  contact,  the  e<|iiat!i>n  '•!    the  conic  is 
''  -''-J  —  "•     <'!assifv    tlie    conic,    jieconlin      to    the    nature    ol'    the 


f,4  T\\t>   DIMENSIONAL  GKOM KTR Y 

S.  Prove  thai  th.'  t  riangle  formed  l>y  the  diagonals  of  any  complete 
quadrangle  who-c  vertices  an-  in  tin-  conic  is  a  self-polar  triangle. 

4.  Prove   ;hat    tin-   triangle   whose   vertices  are   the  diagonal  points 
iif  a  complete   ipiadrilateral  circitniscrilH-d  about  a  conic  is  a  self-polar 
1  rianglc. 

5.  Prove  tiiat   a    range  of  ]ioints  on  any  line  is   projective  with  the 
p.-iu-il  dt  lines  formed  by  the  poiars  oi  t  he  points  with  respect  to  an  v  conic. 

G.  If  /'. .  /'.,,  /'.,  are  three  points  on  a  conic,  prove  that  the  lines  /'_./', 
and  /'  /',  are  harmonic  conjugates  with  respect  to  the  tangent  at  /'.,  and 
the  line  joining  /'._.  to  the  point  of  intersect  ion  of  the  tangents  at  l\  and  /'3. 

7.  If  the  sides  of  a  triangle  pass  through  three  fixed   points  while 
two  of  t  he  vertices  describe  fixed  lines,  prove  that  t  lie  locus  of  the  third 
vertex  is  a  conic. 

8.  The  ei|iiatii'n   j\  -f-  A./',  =  0,   where  J\  and  /,  are  quadratic   pnly- 
nomials    and   A    is   an   arbitrary    parameter,   defines   a  //>•«'•//   <>f  am !<:•*. 
Sketch   the  appearance   of  the  pencil   according  to  the   different  wavs 
in    which   the   conies  f^  =  0  and  f,  =  0  intersect. 

9.  Trove  that   through  an   arbitrary  point  goes   one   and  onlv  one 
conic  of  a  given  pencil  and  that   two  and  onlv  two  conies  of  the  pencil 
are  tanurt'iit  to  an  arbitrary  line.    \\  hat  points  and  lines  are  exceptional  ? 

10.  Show  that  any  straight  line  intersects  a  pencil  of  conies  in  a  set 
of  points  in  involution.    \\  hat  arc  the  fixed  points  of  the  involution  '.' 

11.  Prove    that    the    polars   of   the    same    point  with    respect  to  the 
conies  of  a  pencil  tuna  a  pencil  of  lines. 

12.  If  the  point.  /'  describes  a  straight  line,  prove  that  the  vertex  of 
its  polar  pencil  (  Ex.  1  1  ^  with  respect   to  the  conies  of  a  pencil  describes 
a  conic. 

I'A.  Prove  that  the  locus  of  the  poles  of  a  straight  line  with  respect 
to  the  ci  inics  of  a  pencil  is  a  conic. 

14.  Prove  that  the  conies  of  a  pencil  of  conies  which  intersect  in 
four  di>tinct  points  have  one  and  onlv  one  common  self-polar  triangle. 

ir>.  Pro\e  :  t  tin  pole  of  the  line  at  inlinitv  is  the  center  of  the 
e  conic  is  tangent  to  the  line  at  infinity. 

1  <;.  Prove  that  t  lie  tangents  to  a  cent  ral  conic  at  the  extremities  of  a 
diameter  a  re  i  >:i  ra 

17.  T'.vo  lines  nve  i',:n / Hi/lit,'  with  respect  to  a  conic  if  each  [Kisses 
•:.:<<:_':;  -;."  pole  of  ';e  other.  Prove  that  cadi  of  two  conjugate 
diameters  i-  parallel  to  the  tangents  ;it  the  ends  of  the  other.  Prove 
aKo  that  a  -•>'••  <  '  :  iralle]  chords  are  all  conjugate  to  the  same 

•  'ted  bv  it. 


CUKVKS  OF  SECOND  ORDER   AND   SECOND  CLASS      Go 

18.  Consider  a  pencil  of  lines  with  its  vertex  at  the  center  of  a  conic, 
and  an  involution   in  the  pencil  such   that  corresponding  lines  in   the 
involution   are   conjugate   diameters  of  the   conic.     Show  that  the  fixed 
lines  of  the  involution  are  the  asymptotes. 

19.  The    foe  i   are   defined  as  the   finite   intersections   of  the   tangents 
from  the  eirde  points  at  infinity  to  any  conic.    Show  that  a  real  central 
conic  has  four  foci,  t  \vo  real  and  two  imaginary,  and  that  the  real  foci 
are  those  considered  in  elementary  analytic  ifeomet  rv. 

35.  Classification  of  curves  of  second  order.     \Ve  are  now  ready 
to  find  the  simplest  forms  into  which  the  equation 

2",<-V,  =  0          ("„-"„)  (1) 

can  l»e  put  l>v  a  change  of  coordinates. 
As  before  let  us  place         \ 


CASK  I.  />  —  0.  T]ie  curve  has  no  singular  points  (  sj  '•}'•}  ).  and 
there  can  lie  found  an  infinite  number  of  self-polar  triangles 
(VII,  ^  ->4).  I'Ct  one  such  triangle  be  taken  as  the  triangle  of 
reference.  Then,  since  the  polar  of  0:0:1  is  the  lint1  .r,=  0,  we 
shall  have,  in  the  new  equation  of  the  curve,  //  .,  =  ".,.-"  o.  Since  the 
polar  of  ()  :  1  :  0  is  .ro=  0,  we  shall  have  #,.,=  ".,.=  "•  Since  the  polar 
of  1:0:0  js./-  =  0  we.  shall  have  ci  =  a  —  0.  'I'he  eiiuation  of  the 

1  12  1  •>  1 

curve  is  therefore  ^  +  a^  +  a^=  ()>  (  .-,  } 

\o  one  of  the  coellieients  c/n,  //.,.,,  <>.M  can  be  /.ero,  for  if  it  were 
the  curve  would  have  a  singular  point. 

If  the  coordinates  of  the  original  equation  of  the  curve  arc  real 
and  the  new  coordinates  are  referred  to  a  real  self-polar  triangle 
with  a  real  unit  point,  the  coefficients  ",,,  ".,.,.  and  <i.M  are  real.  \\'e 
may  then  distinguish  two  cases  according  as  all  or  two  of  the  si^ns 
in  (  l!  )  are  alike.  I>v  replacing  ^  '',,  ./',  by  .rt  we  have  then  two 
types  of  equations,  j-,8  +  r.?  +  ./•;  =  0,  (  :',  j 

j-f  +  ./'.;  -  .r:  =  0.  (  1  ) 

'I'he  first  equation  represents  a  curve  with  no  real  points  and 
the  other  represents  one  which  has  real  points.  It  i--  obvious  that 
no  real  substitution  can  reduce  one  equation  i<i  the  other.  <)| 


66  TWO- DIMENSIONAL  (JEOMKTKV 

course  the  second  equation  can  lie  reduced  to  the  first  by  placing 
./•„  /./•,.  which  does  not  involve  imaginary  axes  l»nt  an  imaginary 
value  of  the  constant  /.-..  Summing  up,  we  have  the  theorem: 

.1  i-nri'i-  <>f  xi'i-ii/iil  ordi  r  tr//"Xi-  t'fjimtivn  hux  real  coefficients  «n<l 
u'/ti'-li  /nix  ii"  xi  Hi/alii/'  />"i//t  IK  "in'  iif  tim  tj/jK'x :  an  i may i nary  curcc 
tin1  1'iiiiiiffiin  nt'  H'/u'i'/i  i'ii/i  !»'  rfl  ni-i'il  t<>  the  Jonn  (/>)i  <nt<t  <t  rciil  cnrre 
tin'  1'ijittifimi  "t'  H'/i/i'/i  1'iin  In'  ri'i/iii'i'i/  tn  tin'  form  (4).  If  n<>  occmttit 
I'K  tii/,-1  it  "f  i/iHii/iiHirii'H  flu1  t'l/ttntinn  <>f  <ni>/  curr<>  if  ///,•  xc<'»n<l  order 
d'itli  no  xiiiijiilnr  /'"//it  am  Ac  rcdnct'd  t<>  tJu1  form  (:>). 

('ASK  II.  I>  -' :  0,  hut  not  all  first  minors  of  />  are  /oro.  'J'he 
ciii-ve  has  then  one  and  only  one  singular  point  (  ^  :'>:'>).  This  may 
he  taken  as  the  point  0:0:1.  Then  a  =  a  =  a  =  i).  The  points 
0:1:0  and  1 :  0 :  0  may  be  taken  in  an  infinite  number  of  ways  so 
that  each  is  on  the  polar  of  the  other.  Kach  of  these  polars  passes 
through  H:0:l  (IV,  £ -M  ).  Since  0:1:0  is  the  pole  of  r  =  0  we 
have  n  a  "  in  addition  to  "„..—  0,  as  already  found,  which  is  also 
the  condition  that  1:0:0  is  the  pole  of  ./^  =  0.  The  equation  of 

the  curve  is  therefore  .,  ., 

«irr;  +  ^,,^=0.  (;>) 

Neither  of  the  coefficients  nu  or  a.,,  can  be  /.ero,  for  if  it  were, 
the  curve  would  have  more,  than  one  singular  point. 

Kquation  (  o )  may  be  reduced  without  the  tise  of  imaginary 
quantities  lo  one  of  the  types 

.'Y +./•;-     0,  (I!) 

./Y  -./•;:     0.  (7) 

Summing  up,  \\"e  have  the  theorem: 

^1  I-H/TI'  if  tin'  Ki'i'niii/  on/!'/-  jt'Jtnxc  i'i/iiiif /mi  JKIX  ri'nJ  <'"i'fl/<-iillfx  ninl 
irliii'li  Jut*  "in'  xi iii/iil'ir  jioi nt  ix  inn'  of  lira  II/IH'S:  t/i'ii  i niinji mi r>t  xt ruiiilif 

II in  x   ri  I'i'i  xi  iifi-il  oif  <•</'"'//"/>   ('»)  "/'   firn   fi  a/  xfril/';//lt   //IliX    ri'/'/4i'xi'/l/i'i/ 

III     l-llllllt/'lltl      (7).        If    II"     UlTil/l/lf      IX     (il/,i'/l      if    illllli/lllll  I'll  X      ll      I'll /-''I'     "f 

.s,  ,-,,/!,/  "/-i/i  r  ir/tli  "in'  xhif/lif'tr  }>"iiit  i-"iixixtx  ">'  tim  utrti/i/ht  liinx  inti'r- 
xii-tiin/  in  tjnit  [mint,  n//'/  /'fx  iii/mf i"ti  /inn/  !»'  juit  in  t!n>  f<>rnl  (''). 

('ASI-:  III.    l>    =  0.  and  all  its  lirst  minors  arc /ero.  The  curve  lias 

thru   a   lii f  singular  points,  and  its  c(|Uation   may  be  reduced  to 

.1   ~  (l     (    ^     •)•)).         .1      <•///•''(      i,t     .v,  -i-nilil    iii'ill'l'    ll'lth     it     Ill/I'     "t    XI  t/i/lllil/'    l>"ltltx 

i-.nixixtx   "t'  tli, it    /in.-    f'l/.'i/i    <l>nil,1<\ 


CURVES   OF  SECOND  ORDER   AND   SECOND  CLASS      07 


EXERCISES 


1.  Applv  tlic  foregoing  discussion  to  the  classification  of  curves  in 
Cartesian  coord i  nates,  using  •*'.,=  0  as  t  he  equal  ion  of  t  he  line  at  in  I'm  it  \  . 
Where  does  the  parabola  occur  in  the  discussion  '.'    (See  Ex.  '2,  $  .'>•!.) 

2.  Sho\v    from    the    foregoing   that   if   an   ellipse    or    a    hyperbola   is 

referred   to   a   pair  of   conjugate   diameters,  its  equation   is  =  1, 

a-       ft" 
and  conversely. 

3.  Sho\v  from  the  foregoing  that  if  a  parabola  is  referred  to  a  diam- 
eter *   and  a   tangent    at    the   end   of   the  diameter,   the  equation   of  the 
parabola  is  //~  -  -  cr,  and  conversely, 

4.  Sho\v  that   if  a  central  conic  does  not  pass  through  either  of  the 
circle    point>    at    intinitv,    it,   has   one  and    onlv    one   jiair  of  conjugate 
diameters   which   are   orthogonal    to   each    other. 

5.  Show    that   if  a   parabola,   does    not   pass   through   a  circle   point 
at   infinity   one   and   only   one   pair  of  axes   described   in    Ex.   -1  will   be 
orthogonal.    Write  the  equation  of  a   parabola  tangent    to   the   line  at 
inlinit  v   in  a  circle  point. 

36.   Singular  lines  of  a.  curve  of  second  class.    Consider  the  curve 
of  second  class  defined  by  the  equation  in  line  coordinates 


P>v    ^  '•}-    the    singular    lines    ot    this    locus    are    del'uied    bv    the 
equal  ions 

-'I    ,..",    +    ••'.,..''.,     +-     •'.,;",          =      °'  (-) 

-',.",+     ./.,.,".,+     •'  .,..".,         =      "• 

Let    A,    called    the    iltwftintmtnt    ot    the    curve    (1  ),    be    dctine(l    bv 
the  equation 

A-    .l!.!   .f  Jj 


Tlioro  are  tbeu  tliree  cases  in  the  discussion   of  equations  ("J). 

('ASI-:  I.    A  •'-<).     liquations  (  i! )  have  no  solution,  and  the  curve 
has  no  singular  line.     This   is  the  -general  case. 


OS  TWO-DIM  KNS10NAL  (JKOMKTRY 

CASK  II.  A=0,  but  not  all  the  first  minors  of  A  are  zero. 
Equations  c2)  have  one  solution,  and  the  curve  has  one  singular 
line.  Let  this  line  by  a  change  of  coordinates  be  taken  as  the  line 
0:0:  1.  Tin'  derive  of  the  equation  will  not  be  changed,  but  in 
the  ne\v  equation  we  shall  have  A13  =  An^  =  A.^=  0.  The  equation 

therefore  becomes  ., 

Ant<~  +  -  Al..n}u.,+  A».,u.,=  U, 

which  can  be  factored  into  two  linear  factors.  These  factors  can- 
not be  equal,  for  if  thev  were  we  should  have  A  :  A  ,  =  A  :  .1.,,,  and 
equations  (-).  written  for  the  new  eijuation,  would  have  more  than 
one  solution.  Kach  of  the  factors  of  ('•})  represents  a  pencil  of 
lines  the  vertex  of  which  lies  on  the  line  ./-(=  0  ;  that  is,  on  the 
singular  line  of  the  locus  of  (1).  Equation  (  1  )  is  the  line  equation 
of  the  two  vertices  of  the  pencils  represented,  and  the  singular  line 
is  the  line  connecting  these  two  vertices. 

CASK  III.  A  =  0,  and  all  its  first  minors  are  /.ero.  Any  solution 
of  one  of  the  equations  (-)  is  a  solution  of  the  others,  and  the 
curve  has  a  pencil  of  singular  lines.  If  bv  a  change  of  coordinates 
that  pencil  is  taken  as  the  pencil  »  =  0,  we  shall  have  in  the  new 
equation  (  1  )  A  =  A  =  -•/...,=  A.,3  =  A  ••=  0.  and  the  equation  becomes 
«j"=0.  Hence  in  this  case  equation  (  1  )  is  the  equation  of  two 
coincident  {joints. 

Summing  up.  we  have  the  following  theorem:  A  rttri'i'  <>f  t/n- 
,sv  •"//(/  flit**  liax  m  iji  ni'i'iil  nn  ><i)it/i(I<ir  Inn-.  It  it  J/<ix  <>nt'  tt'inifnlar 
I  tin-  it  rnnxtxtx  "f  t  ii'n  ilist'un-t  jmiittx  /////(//  "//  tJult  line.  If  it  1t<tx  <t 
!„  ,(<•!!  "f  .s////////i//*  /im-x  ft  f/y/.v/x/x  i  if  t/it-  Cfrt<'.r  <  if  tJtnt  jifiifil  <{<>ti/>l// 
f.'.-lc-.n.-.L 

37.  Classification  of  curves  of  second  class.  I>y  §  -\'2  the  limit 
{mints  of  intersect  ion  of  two  lines  of  the  locus 


are  given  bv  the  equations 


At,.it.,+  A 


CURVES  OF  SECOND  ORDER  AND  SECOND  CLASS   (j',1 

CASK  I.  A~  0.  Equations  (-)  can  be  solved  for  n^  //,,  and  ?/.., 
and  the  results  substituted  in  (1).  But  by  aid  of  equations  ('_'), 
equation  (1)  can  be  replaced  bv  the  equation 


The  result  of  the  substitution   is  therefor 


.=  0, 


which  may  be  written  ^''a-'V''/.  --    (l<  (  ~>  ) 

where   <t^_  is   the  cofactor  of  Ja.  in   the    expansion   of   the   dctermi- 
nant    A. 

This  is  the  curve  of  second  class  enveloped  bv  the  lines  which 
satisfy  equation  (1).  It  appears  that  it  is  also  a  curve  of  second 
order.  Let 


it  <!  (I 

i;i          -o          Ja  i 

be  the  discriminant  of  (  ~>  ).    'I'hen 

A      0      0 
D-  A=   0      A      0   =A3 

I  U      U      A 

and  7>--=A-^  <*. 

\Ve  have  therefore  the  following  result:  .1  currc  "f  .scc«////  r/r/.v.v 
//v///  it"  singular  It/if  /.s-  ^/,vo  /<  <-nri'<'  »t'  xn  '«//./  <>,•</,•>•  //•////  //"  Kinijuhir 
ji'iint.  The  converse  theorem  is  easilv  pro\'ed  :  .1  ftirrf  >>/  x,  -•<-//,/ 
n/'ili'/'  icitli  iin  xtnf/nl(tt'  ji'//'nf  /x  (//>•'/  </  i'i//'i'i'  at'  x.  --"//I/  /7'/xx  ^vV/i  y/" 
x/it'/ii/iir  ////(-. 

Since  the  simplest  equations  of  the  curve  ot   second  order  are 


./•f  +./•;-./•;      ". 

the  simplest   e(|Uations  of  the  eur\e  of  second  class  arc 

a  {+  /<:+  it:      ". 

//,'  f   li  :        i<  "       U. 


70  TWO-DIMENSIONAL  (JEOMETRY 

('ASIC  II.  A  I',  but  not  all  its  first  minors  art-  zero.  Equations 
('2)  have  no  solution,  so  that  no  point  equation  can  be  found  for 
the  locus  of  tin-  limit  points  on  the-  lines  of  equation  (1  ).  In  fact, 
we  have  already  seen  that  the  limit  points  are  two  in  number 
only,  the  vertices  of  the  two  pencils  of  lines  defined  by  (1).  The 
simplest  forms  into  which  equation  (1)  can  be  put  without  the 
use  of  imaginary  coordinates  are  obviously 

uf+  i<:=  0, 

I/*-  ,/;=  0. 

CASK  III.    A=  0,  and  nil  lirst  minors  are  equal  to  zero.    We  have 

already  seen  that   the  simplest  form  of  the  equation   in  this  case  is 

u;  =  0. 

38.  Poles  and  polars  with  respect  to  a  curve  of  second  class. 
Equations  (-)-  ^  ;>7,  can  be  used  to  establish  a  relation  between 
any  line  /r,  whether  or  not  it  satisfies  (  1  ),  vj  :>7,  and  a  point  .>\  tie- 
lined  by  these  equations.  The  point  is  called  the  p»li'  of  the  line, 
and  the  line  is  called  the  j><Jnr  of  the  point  with  respect  to  the 
curve  of  second  class  given  by  equation  (1),  §  o7.  The  following 
theorem  is  then  obvious  : 

To  /in//  llni'  i>f  tin'  plitiif  I'orrfuptntdn  a  dixlinrt  j»>l>',  lut  t<>  <tni/ 
point  corresponds  <i  distinct  pn/nr  u-J/i'/i  <in<l  "/////  irhi'ii  tin'  il/xcrt't/i- 
imtnt  nf  tin-  i-nri'f  nf  x,',-unil  cluxx  tlm'x  /i"f  rditixli. 

This  relation  is  dualistic  to  that  of  ^  ->4,  and  all  theorems  of  that 
section  can  be  read  with  a  change  of  "  point  ''  to  "  line,"  "  pole  "  to 
''polar,'  etc.  We  shall  prove  in  tact  that  ///  riixc  uf  <t  I'urve  tif  wnn<1 
urdt'i'  nn<l  xt-coml  r/i/.s-x  irithmit  xitii/H/iir  i><>int  <>r  Hiif  tin-  dt'iihittiina  "/' 
puff*  and  i>nl<ii'x  hi  ^  o4  a/nl  sj  3S  cnfm-idi-. 

This  follows  i  I'oiu  t  he  fact  that  t  he  cur  ve  of  second  class  defined  by 

V.I,////,       I* 

*—/ 

is,  when  A  •'-  0,  the  curve  of  second  order 


win-re  <iik  is  the  eofaetor  of  .liX.  in  A.  Now.  if  equations  (-).  s  :>7, 
are  sol  vet  1  I'm'  ii  ^  a  ...  and  //  ,  there  I'esult  the  etj  nations  (  1  ),  vj  :'>  1,  and 
t  he  t  heoreni  is  i  in  i\  cd. 


CURVES  OF  SECOND  ORDER  AND  SECOND  CLASS      71 

In  case  a  curve  ot  second  class  consists  ot  two  points,  bv  a 
theorem  dualistic  to  IV,  ^  o4,  the  pole  of  any  line  lies  in  the 
singular  line,  which  is  the  line  connecting  the  two  points.  It  mav 
be  found  by  means  of  a  theorem  which  is  dualistic  to  VIII,  ^  )54,  and 
which  mav  be  worded  as  follows: 

//'    it II  If     pn'tnt     M     tX     iitkt'lt     "il      ,1 

li/tt1  i>,  ami  r  ami  x  <in'  tin'  Ituctt 
thruwjh  M  Muni/iny  t<>  a  citrt-i' 
iif'  x,-<-<>n<l  <-/iixx,  ami  <{  ix  tftf  litif 
joiniinj  M  in  tin-  j»,lc  nf  p,  (/,,• 
It/ii'x  i>  iiinl  Y  ''''''  htirinoinc  i'nit- 
jui/ittix  irith  ri'xjh'i't  in  r  ami  x. 

This  theorem  is  illustrated  in  Fig.  l>0,  which  also  suggests 
the  construction  necessary  to  find  /'  the  pole  of  ]>,  since  /'  is  the 
intersection  of  </  and  the  line  <><>'. 


EXERCISES 

1.  If  the  three  vertices   of  a  triangle  move  on  three  fixed  lines  and 
two   of   its  sides   pass  through  fixed  points,  the  third  side  will   envelop 
a  conic. 

2.  A  nuiL,re  of  eiuiies  is  defined  hv  the  equation  _/'+ A/",  =  0,  where 
J\~-  0  and  _/!,-=  0  are  the  equations  in  line  coordinates  of  two  conies. 
Diseuss  the  appearance  of  the  ranp'. 

3.  1'rove  that  there  is  in  general  one  and  oidv  one  conic  of  a  raiiLjv 
which  is   tangent  to  a   ^iven    line  and   two  and  oidv    two  conies   of   a 
ran.ife  which  pass  through  a  ^iven  point.    \Vhat  are  the  exceptional  lines 
and  points  '.' 

4.  Prove   that  for  a  ^iven   ran^e  all   tangents  through  a   fixed    point, 
form  a  pencil  in  involution  with  itself. 

5.  Prove  that,  fora  t^iven  ran^e  of  conies  the  poles  of  a  tixed  Mrai^ht, 
line  form  a  ran^e  of  points. 

G.  If  ;i  straight  lint!  in  Ex.  a  turns  ahoul  a  jioint,  show  that  the  lia^e 
of  tlie  ratine  ot  its  [>olar  points  envelop  a  conic. 

1.  Prove  that  the  centers  of  the  conies  of  a  ran^e  lie  on  a  straight 
line. 

8.  Prove  that  the  conies  of  a  ran^e  with  four  distinct  common 
tangents  have  one  and  onlv  one  self-polar  triangle. 


72 


TWO-DIM  KXSK  )N  A  L  GKOM  KTR  Y 


39.  Projective  properties  of  conies.  We  shall  prove  the  following 
theorems  which  are  connected  with  the  curves  of  second  order  and 
involve  projeetive  pencils  or  ranges. 

/.  '/'//»'  point*  »f  intersection  <>t  correspond/at/  ////ex  "/'  tiro  project  ire 
jn'ticilx  //7//c//  i/"  n<>(  /Hire  {(  cuiinnoii  vertex  </ener<ite  n  curve  of  second 
"filer  ii'Jticlt  piixxes  t/ti'Hlii/h  tin'  vertices  of  the  pencil*. 

Without  loss  of  generality  we  may  take  the  vertices  of  the  two 
projective  pencils  as  A(  0  :  0  :  1  )  and  ('(1:0:0)  (Fig.  'Jl)  respec- 
tively and  mav  take  the  point  of  intersection  of  one  pair  of 


irrcspoiiding  lines  as  />'(0:  1  :  0).    The 
two  pencils  are  then 

./•j +*./-.,=  0 
;uid  sa+\'j-3=Q, 

where  \' -         — -•    The  point /?  lies  on 

y\  +  8 

the    line    of   the    lirst    pencil,    for   whicli 

\  =  0,    and  on    the    line   of   the    second 

pencil,  for  which  \'  =  x.    Since  these  are 

corresponding  lines  in  the  projectivitv, 

we  have  B  =  0.    Then  ft  and  7  cannot  vanish,  owing  to  the  condition 

a&  —  fty  ^-  0.    Now,  if  jc  :  j\t :  ./-.t  is  a  point  on  two  corresponding  lines 

of  the  pencils,  we  have  X=-  — >  X'-       — ,  and  hence 

•ra 
yj-^j'.,  -  fi.i\si\,  +  a./'.j.rj  =0.  (  1  ) 

The  point  j^:  ./•.,:  r.s  therefore  lies  on  a  curve  of  second  order. 

Conversely,  if  //j :  //., :  //.  is  a  point  on  this  curve  of  second  order, 


I>iit    the   line    joining    y.  to  A   has  the   parameter  X=—  '  '.   and 

//„ 

the   line   joining   //.   to   />'   has   the   parameter   \' -         ~  -  and   conse- 
quently  X'  •     Hence   the    point   //,    is    the    intersection    ot 

two  corresponding  lines  of  the  two  projeetive  pencils. 

1  hat    the   curve   ot    second    order   with   the   equation    (1)    passes 
through   ./  ami    ('  U  ol»\ioiis.     Hence  the  theorem  is  pm\ed. 


(TKVKS   OF  SECOND  nKDKK   AND   SECOND  CLASS      To 

If  a  -=-  0  the  curve  (1)   reduces   to  the   two  straight    lines  .'•„=') 
and  JJ\—  $.t'  =  0,  and  the  two  pencils  are  in  perspective  (  sj  1'i). 
Kquation  (1  )  inav  lie  written  in  the  more  symmetrical  1'orm 


//.  T/t>'  hnt'x  count'*-!  nt<j  i'n/'t'f'i<j>mtiltn</  jmt/ifx  <>t  tw<j  }ir»ji'rttvt' 
/v//</r.v  tcJutJi  if"  H"t  It'll',1  the  xiiuii'  l"ix,'  eni't'lun  it  i-nrri-  ut  XCCO/M/ 
rlaxx  u'hich  ix  titiii/i  nt  t<>  thf  Intxi-x  «('  t/t<'  tu'n  riiinjfx. 

This  is  dualistic  to  1.  \\'e  mav  take  the  bases  of  the  two  ranges 
as  n(  0  :  (I  :  1  )  and  r(1:U:(J)  (  l-'jo-.  -2'2)  respectively,  and  a  line 
connecting  two  pairs  of  corresponding  points  as  /M(|:  1  :  ").  The 
line  equations  of  points  on  the 
two  ranges  are  then 
u  +  \nt>—  0 
and  M.(+\'M8=0, 

where,  as  for  I, 

>i\  -f-  tf 
7\ 

'1  he  lines  connecting'  corre- 
sponding points  then  satisfv 
an  ei jiiat ii  1:1  of  the  form 

<•  //  //  -i-  i-  u  a  -f-  i-  a  u  —  1 1. 


(  'oiiverselv.  anv  line  satis!  vin^'  this  eijiiation  is  a  line  connecting 
eoi'i'cspoiid  ni'j;  points  ot  the  two  ranges. 

\Vhen  a  ~  (\  the  eipiation  factors  into  //  —  (>  and  y>/  --  tfu  =  0, 
and  tin-  t\\'o  ranges  are  in  perspective. 

///.    Ai>  if     f"'"    jxitllfx    "II     il     I'll/'i'i      ';/     Kt'i'nUi?     "/•'/'/'    U'ttJli'llt     XtHi/liltlf 

Unfa    until    I,,1    unfit   its    fin'    ri-/'f/i-,'i<    nt    tu-ii    i/<ni/''i///t</   /»  in-ilx. 

No  three  points  ot  the  curve  lie  in  a  straight  line.  Hence  anv 
three  points  on  the  curve  mav  lie  taken  as  the  vertices  ot  the 
coordinate  triangle  AH<\  The  equation  of  the  curve  is  then  of 

the     tol'lll  ,,    ,•    ,-    _j_  ,.    ,-   ,.     j_  ,,    ,.    ,.       -  0,  (    f  ) 

no  siii"'ular  point. 


74 


T  \ V  < ) -J  >  I M  K  N  S I O  N  A  L  (. J  E(. )  M  KT 1 1 V 


The  equation  of  any  line  through  A  is  .^  -f-  X./-i>=  0  and  that  of 
anv  line  through  ("is  .r,+ X'./-.f=  <>.  If  these  lines  intersect  on  (4) 
we  have  ,•  \  —  ,•. 


The  correspondence  of  lilies  of  the  pencil  with  vertex  A  and 
those  of  the  pencil  with  vertex  ('  is  therefore  projective.  This 
j  in  i\  es  I  he  t  heorciii. 

IV.  Anif  tico  t<unjent  I/in  'ft  (D  <(  currc  <if  xeetind  clfixx  without  aim/ulttr 

point*  niiti/  he  tiihen  its  tin'  h<ise#  of  (tt'n  prujectice  yencratiny  ntnyen. 
This  is  dualistie  to  theorem  III. 

V.  If  itni/  point  i >f  it  I'Hi't'e  <>t  nei'<>)nl  order  ivithuut  xini/ular  pointx 
ix  eonni'fteJ  icith  uny  four  points  /'/t  //><•  <-it/-r<-,  tin'  r/-</*.s-  r<<ti»  of  the 
f'*ur  I'o/tni'i'fttii/  ///h'x  /x  <'<>tist<t/tt  _/"/•  ///,•  cnri'i'.     ]t  anij  dnnji'itt  line  to 
it  currf  i >f  Kfcund  flit*.*  icit/i'iiit   xini/nl<ir  /tticx  tx  intcrsi'cti'd  /<//  (///// 
f"iir   tnnii>'iitx,   t/ie   c/v/,vx    rntto   of  the  four  points   of  intersection    its 
constant  for   the   i-nrre. 

4'his  is  a  c-orollarv  to  theorems  III   and    IV. 

VI.  (hie  'tint  on///  n/ii'   riii'i'e   <>t   xi'i-ond  <>r<li  /'  am  In'  p<(stn'<I  ttiromjh 
ti/'i   piiints,  no  Jniir  oj  tchich  /><•  in  <i  xtrnii/ht  line. 

Let  the  live  points  he  //.   /',   /.'.   /,',  and   /,'  (  Fi-jf.  -'•}*). 

l-'roin  /',  which  cannot  he  in  the  same  straight  line  with  /.',  /.',  and 
/,'.  draw  the  lines  /'/',  /,'/!,  /,'/':  and  from  /.'.  which  also  cannot  he 
eollinear  with  /,',  /.',  /,',  draw  /.;/;,  ///',  A'/,'. 
4  hen  there  exists  one  and  oiilv  one  pro- 
jectivitv  (  I.  ^  1  •>  )  hetweeli  the  pencil  \\itll 
vertex  /,'  and  that  with  vertex  /'  in  \\hich 
the  line  /,'/_'  corresponds  to  A'/!,  the  line 

/;A;  to  /,'/;.  and  the.  line  /;/;  to  A;/;.   The 

intersection  oi  corresponding  lines  oi  these     ^' 

project  ive    pencils    determine    a    curve    ol 

second   order  through   the   live  ^iveii  points.    Since  anv  two  points 

on  the  curve  ma  v  he  taken  as  the  vertices  of  the  general  in;_;'  pencils, 

onlv  one  ciii've  can  lie  passet]  through  the  points. 

VII.  <hn-     ilnJ    i.nlil     o/t,'     ,-ni'l'i'     of    xrrninl     flilxs     i-<lll     /"'     <-oH*/ rtlfd'tl 

t<iiii/i'/if    tn    ///•-•    liin-a    ii"  four    nt'   irhii-h    iin'i't    tn    <<   point. 
4'his   is  dualist  ic   to  t  heorcm    \'  I. 


(TKYKS  OK  SKCOND  olJDKK    AND   SKCOM)  CLASS      7.", 


VIII.  Pascal's  theorem.     If  <r   /n-j-<ij/»n   /x   ///*,  •/•//•»'«/   ///  «  ,-n  >•>-,'   <>? 

.XVVM//I/      ii/-i/i  T.      till'     jmintft      "I       I  llttTXi'i't  l<ilt      "t       njij)n>ilti'     ts/i/i'X     III'      t»l      i( 

atrtii'jht  ////••. 

l>v  a  hexagon  is  meant  in  this  theorem  the  straight-line 
figure  formed  by  eonnectinif  in  order  the  six  points  /'.  /.!,  /'.  ]\, 
/.',  /,',  taken  anywhere  on  the  enrve  of  second  order  (  !•  i<_^.  -!)• 
The  opposite  sides  are  then  /,'/.!  and 

/;//,   ./,:/'   and    /•/,;.    /;/;    and    /.;/; 

respectively. 

\\'e  shall  first  assume  that  the  curve 
is  without  singular  points.  Then  the 
points  /,',  /.',  and  /'  do  not  lie  on  a 
straight  line  and  may  be  taken  as  the 
vertices  of  the  triangle  of  reference. 
Let  /;  be  the  point  (<)  :  0  :  1),  /,' 
the  point  (0:1:  <>),  and  //  the  point  ;>l 
(1:0:0).  Then  the  equation  of  the 
eurve.  is,  by  (  "1  ), 


Let     /.!    have     the     coordinates     //., 
/,'     the     ciinrdinates     ?.,     and      /.'     the 

eoi'irdmat es    //•..     Then,    since    the    three    points    /'.     /,'.    and    /,.'    lit 
on    the    curve    (  I  ),    \\c    have 

1         1 


CO 


1          1         1 

//',        //•         ir 


lines    /.,'/,'   and    /.'/.'    intersect    in    the    point    '— '  :        :1    and    the    lines 


TWO-DIMENSIONAL  GEOMETRY 


/'/'  and   /,'/'  intersect  in   the  point   !:—=:—•    The  condition  that 
these  three  points  lie  on  a  straight  line  is 


=  0, 


which  is  readily  seen  to  be  the  same  as  equation  ((I). 

It  the  curve  of  second  order  consists  of  t\vo  intersecting  straight 
lines,  the  theorem  is  still  true,  l>nt  the  proof  needs  modification. 
When  the  points  /',  /.',  and  /.'  lie  on  one  of  the  straight  lines 
and  /.',  74',  /„'  lie  on  the  other,  we  have  the  theorem  of  Pappus 
(VII,  $•>'))•  Other  distributions  of  the  points  on  the  straight 
lines  arc-  trivial. 

IX.  Brianchori1  s  theorem.  Tf«  lif.r<i<i<>n  ?'*  <-ir<'U)n*<-ril>t'<1  <t!»ntf  <i  i-nrrc 
nf  second  cA/xx,  t/n1  lines  connect  ui<f  opposite  vcrtn'cx  nii'i't  in  <>  point. 

This  is  dualistic  to   VIII,  and  the  proof  is  left  to  tin-  student. 


EXERCISES 

1.  Trove   that    the  center  of  hoinology  (see   Ex.0,  $  .">0)  of  two  jiro- 
jective  pencils  of  lines  is  the  intersection  of  the  tangents  at   the  vertices 
of  the  pencils  to  the  conic  generated  bv  the  pencils. 

2.  I'rove   that    the   axis  of   homology  (see    ICx.  10,  :j,")())  of  two   pi-o- 
jcctive  ranges  is  the  line  joining  the  jioints  of  contact  of  the  liases  of 
the  ranges  with  the  conic  generated  bv  the  ranges. 

3.  Show  that  the  lines  drawn  through  a  fixed  jtouit  intersect  a  conic 
in  a  set.  of  points  in  involution,  the  fixed  points  of  the  involution  beni^ 
the  points  of  contact  of  the  tangents  from  the  fixed  point. 

4.  I'rove  that    if    two  triangles  are   inscribed   in   the  same  conic  they 
are  circumscribed  about  another  conic,  and  converselv. 

5.  I'rove  that  if  a  pentagon  is  inscribed  in  a  conic  the  intersections 
of   two  pair>  iif    nonadjaceiit  sides  and   the  intersection  of  the  fifth  side 
and  the  tangent  at  the  opposite  vertex  lie  on  a  straight   line. 

6.  State  and  prove  the  dualistic  theorem  to  Ex.  ~>. 


CTKVKS  OF  SKCON1)  OKDKK   AND   SKCOND  CLASS      77 

7.  Prove  that  if  a  quadrilateral  is  inscribed   in  a   conic  the  inter- 
sections   of    the    opposite   sides    and    of    the    tangents   at    the    opposite 
vertices  lie  on  a  straight   line. 

8.  State  and  prove  the  dualistie  theorem  to  Ex.  <>. 

9.  1!    a   quadrilateral  .{/!''/>   is    inscribed    in   a  conic  and    /.    is   the 
intersection  of  the  tangent  at  .!  and  the  side  li< ' ,  l\   is  the  intersection 
of  the  tangent  at  /.'  and  the  side  A/>,  and  M  is  the  intersection   of  the 
sides  A/:  and  ''/>.  prove  that  /.,  K,  and  M  lie  on  a  straight  line. 

10.  State  and  prove  the  dualistie  theorem  to  Kx.  S. 

11.  I  f  a  t  riangle  is  inscribed  in  a  conic,  prove  that  the  intersections  of 
the  tangents  at  the  vertices  with  the  opposite  sides  lie  on  a  straight  line. 

12.  State  and  prove  the  dualistie  theorem  to  Kx.  1-. 

13.  Prove   that  the   complete  quadrangle   formed  by   four   points  of 
a    conic    has.    as    diagonal    points,    the    points    of    intersection    of    the 
diagonal    lines   of  the   complete   quadrilateral    formed    bv  the  tangents 
at  the   vertices   of  the  complete   quadrangle. 


CHAPTER  VI 

LINEAR  TRANSFORMATIONS 

40.  Collineations.    A    collineation    in    a   plane   is    a   point   trans- 
formation (J5  f>  )  expressed  by  the  equations 


(  1  ) 


If  the  determinant  VA    is  not  equal  to  /ero,  these  equations  can 
be  solved  for  r,  with  the  result 


(2) 


where  A<k  is  the  cofaetor  of  nik  in  the  expansion  of    ^    and  where 
[,<„  -0. 

If  the  determinant  j"a.[=0,  equations  (2)  cannot  he  obtained 
from  (1).  For  this  reason  it  is  necessary  to  divide  collineations 
into  two  classes  : 

1.    Xonsingular  collineations,  for  which  ]aik.  ^  0. 
'2.   Singular  collineations,  for  which    <itl.  =  0. 

\Vc  shall  consider  only  nonsin^ular  eollineations  in  this  text, 
though  some  examples  of  singular  collineations  will  be  found  in 
the  exercises. 

It  is  obvious  that  for  a  nonsingular  eollineation  rt  cannot  have 
such  values  in  (  1  )  that  ./•(  —  r.'  =  j-'3--=  0.  Hence  bv  (1  )  anv  point./; 
is  transformed  into  a  unique  point  .r'r  Similarlv.  fVom  ('2)  any 
point  ./•'  is  the  t  ransformed  jioint  of  a  uni»|iie  point  ./\. 

Consider  now  a  straight   line  with  the  LMJ  nation 

;/i.ri-(-  ;/,.r,  -f  ",-/•.,=  0. 


LINKAK   TRANSFORMATIONS  711 

All  points  r,  which  satisfy  this  equation,  will  be  transformed  into 
points  j-[,  which  satisfy  the  equation 

ntf +  i/X  +  "X=o1 

where,  by  ('2), 


It  appears  then  that  any  straight  line  with  coordinates  */,  is 
transformed  by  (1)  into  a  unique  line  with  coordinates  //.  Also, 
equations  (3)  may  be  solved  for  ",  with  the  result 

-"-.X+'vJ, 


\>t,=  If,  .,11 


4- 


from  which  it  appears  that  any  line  is  the  transformed  line  of  a 
unique  line. 

Equations  (8)  express  in  line  coordinates  the  same  transforma- 
tion that  is  expressed  by  equations  (1)  in  point  coordinates.  For 
it  is  easy  to  see  that  by  equations  (-5)  any  pencil  of  lines  with  the 
vertex  r,  is  transformed  into  a  pencil  of  lines  with  the  vertex  ./.  and 
that  the  relation  between  .r  and  .r'  is  exactly  that  given  bv  equa- 
tions (  1  ).  Eq  nations  (  •>  ),  therefore,  which  express  a  transformation 
of  straight  lines  into  straight  lines,  also  afford  a  transformation  of 
points  into  points  in  a  sense  dualistic  to  that  in  which  equations  (1  ) 
afford  a  transformation  of  straight  lines  into  straight  lines. 

We  will  sum  up  the  results  thus  far  obtained  in  the  following 
theorem  : 

I.  />'//  a  )>»)ix/)///i//<ir  <'<>lli)icrttii»l  in  ii  jiliith-  i'r,'rij  )><>hit  fx  trinix- 
fufnif'il  hif'i  <i  unii^ne  i»>i>it  ini</  >•  >;•/•!/  xtruii/ht  Inn-  //if"  *i  tnn<jii,' 
nfrtiit/Jit  Jin  i''  itrni,  ivwrprm'lifi  t'l'i-ri/  >i«>/if  /x  ///»•  twimtf'iriHi'il  /»>/'iif  »>'  «? 

in>-  f/n-  trtnisnrw<'<I  tin>'  »    •<  iini<><>' 


('misider  now  a  eollineat  ion    I!    by  which   any   point    .i\  is  trans- 
formed  into  the  point  ./,  where 

p.r(  —  nnJ'}  +  ",-•'':+  ",.r^i' 

and  let   //.,  be  a  collineatiou  by  which  any  point   ./•[  is  transformed 
into  .r",  where  ,,      ,       ,       ,       ,       , 


80  TWO  DIMENSIONAL  (JKOMKTKV 

Thru  tin1  product   A'.,  A'    i*  a  substitution  of  the  form 

Tf"     --  '',,./•,  +  <',•.,'•••+'•,:.•':.' 

which  is  ;i  collineation.    Hence  the  product  of  two  collhieations  is 
;i  collineat  ion. 

Moreover,   it   //    is  as  above  and   A',  is  of  the  form 


the  product   A'  A'    is        .. 

r.r    =  .r 


which  is  the  identical  substitution.  Hence  in  this  case  A',  is  the 
inverse  substitution  to  A^  and  is  denoted  by  A',  '.  Our  work  shows 
that  the  inverse  transformation  to  a  collineat  ion  always  exists  and 
is  itself  a  collineat  ion. 

These  considerations  prove  the  following  theorem: 

//.   The  totality  »f  HdHxinj/iilar  collint'fitinnn  t/i  <i  }>/</>/<•  form  <i  <;r«ni>. 

\Ve  shall  now  prove  the  following  theorems: 

///.  //'  /,'.  A!,  /,'.  /,'  ore  <nii/  f<>nr  itroitrarit  ij  axxunti'il  pointx,  n»  three 

of  u'hli'h   iii'i'   on    tin'  x/tnic   Htrtit'i/ht  line,   <nnl   ![',    /.!',    /.'',    /,''   are   nlxn 

t'niir  it  rl>it  raril  ii  (iMiunctl  pointz,   no  three  of  u'h'fJi   lie  mi  <t  xtrniijht 

line,  f/iere  e.risfx  one  an*!  null/  one  auUineation  /'//  nieunx  of  irlii<'}i   /j  ix 

transformed  into  /,'',   /'  info  /._!',  /_'  Into  /.!',  </n<l  74*  into  /4''. 

To  prove  this  we  will  first  show  that  one  and  onlv  one  eollinea- 
tioii  exists  which  transforms  the  four  fundamental  points  of  the 
coordinate  system,  namelv  ./  (0  :  0  :  1  ),  /*  (0  :  1  :  0  ),  f'(  1  ;  0  ;  0  ),  ami 
/  (1:1:1),  respect  ivelv,  into  four  arbitrarv  points  /'  (  a,  :  a.,  :  n.,  ), 
/.'  (  /3{  :  ft..  :  fi..  ).  /.;  (  7,  :  7.,  :  7,,  ),  and  I\  (  5,  :  &.,  :  8,  ),  no  three  of  which 
lie  on  a  straight  line. 

liv  substituting  in  equation  (1)  the  coordinates  of  correspond- 
ing points,  remembering  that  the  factor  p  mav  have  different  values 
for  different  pairs  of  points,  we  have  the  following  equations  (|ut 
ot  which  to  determine  the  coefficients  <iit: 


Pi'1--     '"•::: 


p  ?>   =  n     -f 

"l      I  11      ' 


LINKAK   TKANSFOK.MAT10NS  Si 

1>\   substitution  from  e^uat  ions  (•">)  in  equations  (ti)  we  have 

p  'i   -f-  p  rf  -\-  p  7  —  p  6  =  0, 
r  i    i    '    i  -'   i   '    r  ;  1  1       "41 


which  mav  IK-  solved  for  p  :  p,:  p.:  p4.  Since  no  three  of  tin-  points 
/'.  /!.  /'.  /4'  lit-  on  a  straight  line  no  dflcnninant  of  tin-  third 
order  funned  from  the  matrix 


can  vanish,  and  hence  no  one  of  the  factors  p.  can  he  zero.  The 
values  of  p  .  p,.  p..  and  p(  luivilljjf  thus  hecii  deterniiiied  exee[>t  for 
a  constant  factor,  tin.-  values  of  the  coefficients  iiit  can  he  found 
from  (.))  except  for  this  same  factor.  Hence  the  collineatioii  (1) 
is  uniijiiely  determined,  since  onlv  the  ratios  of  <iti  in  (1)  are 
e^selit  lal. 

Let  it  lio\v  he  required  to  t  I'ansfol'lii  the  four  points  /,',  /.'.  /..'.  /4'. 
no  three  of  \\hich  are  on  a  straight  line,  into  the  four  points  /,'', 
/.!'.  /'',  I',',  respect  i\cl\'.  no  three  of  which  are  on  a  straight  line. 
As  we  have  seen,  there  is  a  iini(|Ue  collincution  //  \vliicli  transtonns 
.1.  //..  (\  I  into  /,',  /',.  /'.  /('  respi'divelv,  and  a  unique  cnllinoa- 
t  ion  I,'  which  transforms  ./.  />',  (\  1  into  /,''.  /!'.  /''.  I\'  rcspecti\'el\'. 
Then  the  col luicat ion  7,^  '  (theorem  II)  exi>ts  and  tran^tornis 
/;.  /'.  /;.  /;  into  ./,  /'.  <',  1  resjiectively.  The  product  //  //;  '  is 
a  colliiieation  (theorem  II)  \\hich  transforms  /.'.  /'.  /.'.  I\  into 
/,''.  /.:'.  /''.  /,''  respectively.  Moreover,  this  is  the  onlv  collineatioii 
\\hich  makes  the  desired  transformation.  Fur  let  /.'  he  a  collinea- 
tioii which  does  so.  Then  /,',  '  /,'  transforms  /,'.  /',  /'.  /,'  into 

.1.     /•',     ( ',     I    respect  JVelv.       Ilellce 


Thi-  estahli-hcs  the  theorem.  It  i-  inn  |H'eessar\  that  all  the 
points  /;.  /'.  /;.  /;  should  he  distinct  from  the  points  /.''.  /''.  /:'.  /;'. 
In  the  special  case  in  \\hieh  /,'  is  the  same  a^  /''.  /'  the  same  as 


S*J  T\YO-J)I.  MANSION  A  L  (!  KOMKTRY 

/!',  7.J  the  same  as  /!',  and  /,'  the  saint1  as  74'',  7^  =  7i'o  and  7t'  is 
tin-  identical  substitution.  Ilt-nce  we  have  as  a  corollary  to  the 
above  theorem: 

IV.  Anif  i'i>/fini'iif/'"H  tr/f/i  _  fun,'  fj.n'd  jx'intx  tin  tJo'i'c  of  which  nre  In 

tin'  xilnh'  Ktl'rtii/flt  Inii'  /x  tin'    'nl,  n/ii'il/  xn/ixf/fiiti'Hi. 

V.  A»>f   nnnxinijuliir  coll  incut  inn   entalilixhcs  <t  project  ivity  between 
ttic  points  if  tic<>  correspond  !n<f  r<tni/ex  and  t/i>'  line*  <f  tim  corrt'upnnd- 
in<!  I'eiK'ilx,  mid  <ni//  xi«-h  productivity  }»«//  Ic  established  in  an  infinite 

<>    iriii/>*  I//  <i  nnnsini/ulur  coUincation. 


'I'o  prove  the  first  part  of  the  theorem  let  the  pointy.  l>e  trans- 
fonned  into  >/'{  and  the  jioint  -r,  be  transformed  into  z(  by  the  collinea- 
tion  (  1  ),  so  that  , 

—  ' 


,  ,-11  ,-3,  i  83' 

Then    /+  A.?.  is  transformed  into      ,  \\hcre 


whence  ( 

where  X'=- 

P, 

This  establishes  a  projeetivity  between  the  jioints  of  the  range 
//,.  +  X^(.  and  those  of  the  range  //[  +  XV.  \\\~  tin-  use  of  line  coordinates 
and  ci  juat  ions  (  '•>  )  the  proof  mav  bi-  repeated  for  the  lines  of  a  pencil. 

To  prove  that  there  are  an  infinite  number  of  nonsingular  col- 
lineations  which  establish  a  given  projectivity  between  the  points 
of  t\\'o  ranges,  it  is  onlv  necessary  to  sho\v  that  there  are  an  infinite 
number  of  collineations  which  transform  anv  three  points  7',  (J,  R 
Iving  on  a  straight  line  into  anv  three  points  7''.  (,>',  7<'',  also  on  a 
straight  line,  and  applv  III,  vj  1  ">. 

To  prove  this,  draw  through  />  any  straight  line  and  take  ,V  and 
'/'  t\vo  points  on  it.  Draw  also  through  Ji'  any  straight  line  and 
take  .s"  and  7''  anv  two  points  on  it. 

Then  bv  theorem  III  there  exists  a  eollineat  ion  which  trans- 
forms the  four  points  /'.  o,  ,s',  '/'  into  the  four  points  7'',  (/,  S',  T', 
and  this  eollineat  ion  transforms  //  into  /,''.  Since  ,S',  7'  and  .V',  T' 
are  to  a  lar_n-  extent  arbitrary,  tliere  ai'e  an  infinite  number  of 
requ  ired  eollineat  ions. 


L1NEAK   TKANSFOKMATKLNS 


S3 


If  it  is  required  to  determine  a  collineation  which  establishes  a 
projectivity  between  two  <_riven  pencils  ot  lines,  this  niav  he  done 
bv  establishing  a  project  i\  ity  between  two  ranges,  each  of  \\hich 
is  in  perspective  with  one  of  the  pencils.  Since  this  niav  be  done 
in  an  infinite  number  of  ways,  there  are  an  infinite  number  of  the 
required  collineations. 

41.  Types  of  nonsingular  collineations.  A  coll  ineat  ion  luus  a  //./r</ 
[mint  when  ./•/  —  .r,  in  (-([nations  (1),  ^  I".  The  fixed  points  are 
therefore  given  by  the  equations 

(«„-  P  )•'•,+  ",,•'•;  -•-",/:«  =0» 


The  necessary  and  snilicient  conditions  that  these  equations  have 
a  solution  is  that  p  should  satisfy  the  equation 


Similarly,   the   fixed   lines   of    the   collineation   are 
equations 

"~  ~  ° 


(•_') 

b      the 


and    the    necessary   and    sufficient    condition    that   these    equations 
have   a  solution   is 


(I) 


Equations  (-)  and  (-1)  are  the  same  and  will  be  written 


Now  let  p^  be  a  root  of  (."">).     Then  f. 
hypothesis    (tlk  j  <•'-  0.    The  rout  p^   is  a  double  mot  win 


,,  '(t   —p    <i 

t  (p  \~         -^     ^i     -•'  11 

''^  ",,    'P,  ",;, 

and  it  is  a  triple  ro<>t   when 

rcpt)=-j  [(%-/>,) 


S4  TWO-DIMENSIONAL  (JEOMETKY 

We  mav  now  distinguish  three  cases: 

1.  When  all  the  lirst  minors  of  the  determinant  ^(/j  )  do  not  vanish. 
Equations  (  1  )  and  ( -I )  have  each  a  single  solution.  The  eollineation 
has  then  a  single  fixed  point  and  a  single  fixed  line  corresponding  to 
the  value  pr  The  root  p}  may  he  a  simple,  a  douhle,  or  a  triple  root 
of  (;">),  according  as  equations  (i!)  and  (7)  are  or  are  not  satislied. 

'2.  When  all  the  lirst  minors  of./'f^)  vanish,  hut  not  all  the 
second  minors  vanish.  Equations  (  1  )  and  ('-\ )  contain  then  a  single 
independent  equation.  The  eollineation  has  then  a  line  of  lixed 
points  and  a  pencil  of  lixed  lines  corresponding  to  the  value  p^. 

The  root  pl  is  at  least  a  douhle  root  of  (•>)  since  equation  (t!)  is 
necessarily  satisfied,  and  it  may  or  may  not  he  a  triple  root. 

o.  When  all  the  second  minors  of  ./'(/>)  vanish.  Equations  (1) 
and  ( •))  are  satisfied  hy  all  values  of  .r,  and  n.  respectively,  and 
the  eollineation  leaves  all  points  and  lines  fixed.  The  root  p  is  then 
a  triple  root  of  ( •"> )  since  equations  (li)  and  (7)  are  satisfied. 

From  this  it  follows  that  <i  i-allini'ittinn  Ji<is  ax  nnmi/  //./v/  l/'/n-x  KX 
//./>'</  point*  iiinl  its  ntnnif  pencil*  "/  fij'cil  Inn'*  <ix  Inn's  of  fl.i'i'</  points. 

From  £  \'l  it  follows  also  that  ///  c/v/y/  //./•*•</  lint1  lies  <tt  least  one 
fl.iiil  point  inn/  f/i'if  t/i/'oni//i  fViTtj  //./>-»/  point  </<n'x  <it  li'/ixf  on<'  li.n'ij 
lim  .  Tin'  lini'  i'"ii m  <•( tioj  t/ro  fi.t't'i/  points  is  //./•»•</  itinl  tin'  point  i-utn)it<>n 
to  fi./i'i/  tint's  is  (l.i'i'iJ. 

\\'e  are  now  prepared  to  classify  collineations  according  to  their 
fixed  points  and  to  give  the  simplest  form  to  which  the  equations 
of  each  type  mav  he  reduced.  We  will  first  notice,  however,  that 
if  the  point  .rf  =  0,  ./-.  =  0,  .rt  =  1  is  fixed,  then  hy  (1),  £40, 
(iit=  iijk=  0  ;  and  if  the  line  .i\. -•-  0  is  fixed,  then  <tu  =  <<ki  ---  0. 

.1.  Ci'lliiH-ntinHK  n'ith  nt  /xist  //,,;>'  ti.ru/  points  not  in  tin'  s,t//i>' 
sf/'dii//if  'inf.  Take  the  fixed  points  as  the  vertices  J,  /A  ('  of  the 
triangle  of  reference.  Then  the  eollineation  is 

p.i\        "n.'P 
p.l-',  -  -  '/ /•„ 


LINK  A  K   TRANSFORM  ATI*  >NS 

TYI'K    I.  p.r{  =  '/./',, 


The    collineation    has    only    tin-    fixed    points    .1,    //,    ('   and    the 
fixed   linrs   .I//,    />V,  and    '7'. 

Tvi'K    II.  p.i\        «.rr 

p.i''.  —         '/./'.,, 

P'.'  :  :  '-''a- 

The  eollineation  has  the  lixed  jioint  .1.  the  line  of  lixed  points 
/•''',  the  fixed  line  I!C,  and  the  peiieil  of  lixed  lines  with  vertex  .  I. 
It  is  ealled  a  li">//ol"<///. 

TYI'K    III.  p.r{  -./'„ 


p.r.  =          J's. 
All  points  and  lines  are  lixed.    It  is  the  identical  transformation. 

//"/  in  t/n'  xi/iiif  xtrnii/tit  line.  \\  e  will  take  the  two  lixed  points 
as  .1  (11:0:1)  and  <  '  (  1  :  <>  :  <» )  of  the  triangle  of  reference.  The 
collineatioii  has  at  least  two  distinct  lixed  lines  one  of  which  is  AC. 
The  other  must  contain  one  of  the  fixed  points,  and  we  will  take 
it  as  lie  (./-._-  0).  The  eollineation  is  then 

pj\  =   «,,.'•,+  "„.'.,, 
p. >''.,  -  (/.,.,.>'.,, 

P-'a  =  ":;,r'V 

Here  '/  -  "  or  we  should  have  case  ./.  \Ve  shall  place  <7  , -=  1 . 
The  equation  <  .">  >  is  no\\"  (<(  —p)(tiit  -  p  )('/.,-  • /D  ) --  (t.  Placing 
/j  ",,  \\  e  have  as  the  equations  to  dcterniine  the  corresponding 

lixed  point 

(",,-0-'-,+  .'-,=  ^ 

(»    —  ".,,  )•>'..=  ()- 
Since    1>\     hvpnthesis    everv    fixed    point    lies    on    ./•     -  0.    \\  e    have 

'/u  '',,.  It  is  left  Ulldeterillilleil  \\  het  her  " '..is  or  i>  Hot  eijlial  to  ,;__. 
I  lellce  \\  e  ha\  e  t\\'o  lleW  t  V|  M'S. 


SG  TWO-DIMENSIONAL  GEOMETRY 

Tvi'K  IV.  p.r[  =  <u\+  J:,, 


Tin-  eollineation  has  unly  the  fixed  points  A  and  (.'  and  the 
lixed  lines  .1  ('  and  lit '. 

TVIM:  \".  p.t-(  =  tu\  +  i-., 

p/,  =  tu-.,, 

Pa  =  "JV 

'I  he  eollineation  has  the  line  of  lixed  points  AC  and  the  pencil 
of  lixed  lines  \vith  its  vertex  at  < '. 

In  either  Type  I Y  or  Y  the  point  /•'  may  he  taken  at  pleasure 
on  the  line  /•'''. 

('.  <  '"///in 'dti'inx  tc/'tJi  ">tli/  fine  fixed  j>»ittt.  Take  the  fixed  point  as 
<'  (1:0:0).  Tin-  eollineation  has  also  a  fixed  line  whieh  must 
pass  through  < '.  Take  it  as  1>C  (j-  =0).  The  eollineation  is  now 

i  o  \     3  -^ 

p.r(  =  uuj\+  «,,.'•,+  <*13r3, 
p.f',=  a.,,j\^  a.,.^s, 


Equation  ( .") )  is  now  (  «  —  p  )  ( </.,.,—  /?)  ( </  ...,—  p  )  =  0,  and  since 
bv  hypotheses  ('  is  the  only  lixed  point,  we  have  (in  =''.>.,— <'a.,- 
The  point  A  (0:0:1)  taken  at  pleasure  is  transformed  into 
A'  ( -/^  :  rr, .:</.,),  and  if  we  take  the  line  A  A'  as  j^  =  0,  we  have 
ti=().  The  eoeftieients  a  ,  and  <;,  cannot  vanish  or  \\'e  have  the 

previous  cases.     We  mav   accordingly   replace   j\t  by  -  "L  and   j'3  by 
and  have,  tinally, 

K  VI.*  p-;  -,/./-!  4-  r,, 


LIN  FAR    TRANSFORMATIONS  S7 

EXERCISES 

1.  Find  tlic  fixed  ]>i>mts  and  determine  tin-  type  of  rollint'iltion  to 
which  each  of  the  following  t  I'ansi'orniat  ions  in  ('artesian  coordinates 
U-long  :  (</)  ;i  translation,  i  i>  )  a  rotation  ;il>out  a  tixed  point,  i  <•  i  a  reflect  ion 
on  a  straight  line. 

'2.  hetcrmine  the  group  of  eollineat  ions  in  ('artesian  coordinates 
which  leaves  the  pair  of  st  raight  lines  j:~—  i/~  —  0  invariant  and  discuss 
the  subgroups. 

3.  Are  t  \vo  eollineat  ions  with  the  same   fixed  points  always  commu- 
tative'.'    Answer  for  each  type. 

4.  ('onsider  the  singular  eollineat  ions.    Trove  that  there  is  always  a 
point  or  a  line   of  points   for  which    the  transformed   point    is  indeter- 
minate.   "\Vc   shall   call    this   the   singular   point   or   line.     If   there    is   a 
singular  point,  every  other  point  is  transformed,  into  a  point  on  a  lixeil 
line   which  mav  or  may   not  pass  through   tin;  singular  point.     If  there 
is  a  singular  line,   every   point    not   on    the   line    is   transformed  into  a 
fixed  point  which  may  or  may  not  lie  on  the  singular  line.     J'rove  these 
facts  and  from  them  show  that  the  singular  eollineat  ions  consist  of  the 
following  t  \  pcs  : 

I.    One   singular   point    /',    a    fixed    line  j»   not   through  /',    two   tixed 

points   on   /i.  , 

ps\=  •''.> 

p.c',    -         (/./'.,, 


IT.    One    singular    ]ioint    /',    a    fixed    line    not    through    /',    one    tixed 
point  on  n. 


P-'-     -. 

I  \'.    ( >ne  singular  point  /',  a  fixed  line  />  t  h  rough  /',  one  jioint  of/,  fixed. 

P-''\  •>'.',, 


88  TWO-  DIMENSIONAL  GEOMKTHY 

V.   One  singular  point  /',  a  fixed  line/;  through  /',  no  point  of  j>  lixed. 

p-i'i  =       •>"•>, 
p.r'2  =  xa, 

P-''a  —  ()- 
\"I.    A  singular  line  j>,  a  fixed  jtoint  /'  on  p. 


\'ll.    A  singular  line  j/,  a  fixed  point/'  not  on  j/. 


42.  Correlations.    The  equations 

K='Vi+"  ,/.+  "i/a< 

K=  Vi+Va*  Vs»  C1) 

/><==";H-/"l+'V,+  <W 

\\-here  ./•.  are  point  coordinates  and  M'  are  line  coordinates,  define 
a  transformation  of  a  point  into  a  line.  Such  a  transformation  is 
called  a  cnrrflntinn.  As  in  the  ease  of  eollineations,  we  shall  dis- 
tinguish between  noiisingular  and  singular  correlations  according 
as  the  determinant  aik  does  not  or  does  vanish,  and  shall  consider 
only  noiisingular  correlations.  Equations  (1  )  can  then  be  solved  for 
j;  with  the  result  ,  ,  ,  .  , 


where  .li(l.  is   the  cofactor  of  </a.  in   the   determinant     <iil:\.     Every 
straight   line  ut'  is  therefore  the  transformed  element  of  a  point  ./•,'• 
Consider  now  the  points  of  a  line  given  bv  the  equation 

ii  j-  4-  '/.,-''.,  -4-  ^...r,=  *K 

\\here   ni  are   constants.     l>y(-)  these    points  go   into  a  pencil   of 
lines  the  vertex   of  which  is  the  point  ./•,',  where 


LINKAK   TUANSFOJLMATJONS  SO 

We  mav  express  this  by  saying  tluit  tht-  line  n{  is  transformed 
into  the  point  ./•/.  Also,  since  equations  (_'-})  can  be  solved  for  ?/. 
with  the  result 


every  point   is  the  transformed   element   ot'  one  and   only  one   line. 
Since  equations  (-!),  (  •>  ),  and  (4)  are  consequences  of  equations 
(  1  ),   \ve   shall   consider   them   as   given    with  (  1  )  and   sum   up  our 
results    in    the  following   theorem: 

7.  .1  n»nxini/i(l(ii'  correlation  defitu'd  l<y  equations  (1  )  ix  a  trnnx- 
J  orinnt  ton  by  tt'liK'h  etich  point  tx  t  rdnxj  onned  into  <i  tttfiliyht  It/if  ttml. 
t-iii-li  xtt'dii/ht  ll/ti'  int<>  ,i  jioint,  in  xucJi  it  i/niiun:r  t/mt  point*  trhii'h  lie  on 
n  Kt/'itlt/ht  din'  arc  tranxfurnu'd  into  #trai</Jtt  /inrx  n'hi<-h  f><txx  t///-"i///h  <i 
l"i//if.  iiii'i  Inii'x  ii'liii'h  ]»(xx  t/i/'on<//t  <(  point  <i/'f  t  ninxlornii'd  into  jiotntx 
H'hii'li  lii'  o,t  n  xfr<ti'j}it  lint'.  PJtich  lint1  »r  point  ix  transformed  into  nni- 
point  of  li/ti-  n.nil  ix  tin-  t  rit  nxt  ofnu'il  element  oj  <>ne  line  <>r  point. 

('oiisider  now  a  correlation  S  by  which  a  point  .r  is  transformed 
into  a  line  ///,  and  let  N,  be  a  correlation  by  which  the  line  //,'  is 
traiisforuu'd  into  a  point  ./'.  It  is  clear  that  the  product  .S'.N  is  a 
linear  transformation  by  which  the  point  j\  is  transformed  into  the 
point  ./•'':  that  is,  a  collineation.  Therefore  the  correlations  do  not 
form  a;^Toup.  It  is  evident,  however,  that  the  inverse  transformation 
of  any  correlation  exists  and  is  a  correlation. 

We  can   therefore  prove  the  following  theorems: 

77.  If  /,',  /.;.  /.'.  /,'  nre  four  art>it  r<tri/  pointx,  no  tJiree  of  which  lie 
on  a  xt/'iu'r/ht  I!  ni'.  iiml  if  p  .  p  t.  //.,  p  nre  four  nrl>it  ntri/  linex,  no  tli/'-'e 
oj  ii'lii'-li  piixx  tliroiii/Ji  it  point,  thi'/'e  e.rix/x  one  itntl  "/////  on,'  ,•,,/•/>•/,/- 
tion  /,//  ni,-,i)ix  if  irfiirl,  /'  ix  Iritnxfonni'il  into  jt  7',  into  p  ,  /'  into  p  , 
<ii('l  /I  into  p  ,  <t  ml  ///<'/v  i-.iixtx  it/xn  one  a/i'l  <>nli/  one  correlittinH  /-// 
noiinx  oj  n'hi'-h  p  tx  trttnxfnrittetl  into  /J,  pt  into  l'i%  p  into  /.^  and 

/',  in*"  /.;• 

777.  An//  nonxi  iii/nlii  r  cnllineitttnn  fxtitfifixht'n  n  pr<>jeetivitii  Intn-^n 
t  h>  pointx  ut  it  ritnife  iind  tin'  ////ix  of  n  I'ori't'Xjionil  /  it'i  pftc'il.  <//<</  im  // 
xi/f/i  project  it'lttf  until  o>  rttfiltilixfieil  in  <n<  infinite  nunJ«-r  »J  i/'<iijs  /'// 
'/  eorrelot'uni. 


00  TWO  DIMKNSioKAL  CiHO.MKTRY 

'I'ht'  proofs  of  these  tlieoreius  ;iro  the  same  as  those  of  the  cor- 
ivsnoiulimr  theorems  of  i;  1<>  ami   need  not   he  repeated. 

1  r~>  >  i 

\\\  equations  (  1  )  a  point  ./,  lies  on  the  line  u',  into  which  it  is 
transformed  when  and  only  when 

"n-'V  +  ".-/I  +  'Va  +  <  "u  +  ",.  >  Va  +  (  "i-J  +  ".n  >'V'3 

+  C'',.i,+  t'3..)-V'a  =  °'  (;"') 

That    is,  .r    lies  on  a  eonie    A"(. 

Similarlv,  from  equations  (^5)  a  line  M;  passes  through  the  point 
.r  .  into  which  it   is  transformed  when  and  only  when 


4-(.-/ga+J32)jyia=0.  («•>) 

That  is,  M(.  envelops  a  eonie  A',. 

It  is  evident  that  the  conies  J\}  and  A",  are  not  in  general  the 
same.  Their  exact  relations  to  each  other  will  be  determined  later 
in  this  section.  In  the  meantime  we  state  the  above  result  in 
the  following  theorem  : 

IV.  In  tin1  nixe  <>f  (i/i//  n<mi>in</uhir  roi')'<'I<iti<>n  tin-  jmiittx  trJtt'iJi  He 
i>n  t/«  ir  t  t'lUixfurmed  lan-x  (//•<•  fxnntx  «t  a  ci'rttttn  <-<»tt<',  «/td  t/te  lines 
tt'li'fh  IKIXX  tliromjli  thi'tr  tmnxfut'ini'd  point*  envelop  <t  ccrtdi/t  <'u/iicy 
ti'lii/'/i,  in  i/i'iii'i'itl,  tx  not  tlic  xitnu'  «t>  tJtc  firxt, 

Anv  point  /'  of  the  plane  mav  be  considered  in  a  twofold  manner  : 
as  either  an  original  point  which  is  transformed  bv  the  correlation 
into  a  line  or  as  a  transformed  point  obtained  from  an  original 
line.  It  /'  is  an  original  point  it  corresponds  to  a  line  p'  whose 
coordinates  are  given  by  (1  ).  If  /'  is  a  transformed  point  it  corre- 
sponds to  a  line  />  whose  coordinates  are  given  by  (4),  in  which  we 
must  replace  ./  bv  ->',.,  the  coordinates  of  /'. 

The  lines  />  and  //do  not  in  general  coincide.  When  they  do 
t  he  line  1  1  and  the  point  /'  are  called  a  Jmihl,-  jniir  of  the  cor  re  lat  ion. 
That  /'  should  be  a  point  of  a  double  pair  it  is  nccessarv  and  suffi- 
cient that  t  li>'  coordinates  n\  and  nt  of  equal  ions  (  1  )  and  (1  )  should 

be  proportional;   that  is,  that   the  coordinates  of  /'  should  satisfy 
ii  ^ 

the  equations 


I.INKAR   TRANSFORMATIONS  HI 

where  p  is  an  unknown  (actor.  For  these  equations  to  have  a  solu- 
tion it  is  necessary  and  sut'liciciit  that  p  should  satisfy  the  equation 
.  —  on..  "..  -  P" ..  «...  pit. 


The  correlations  niav  be  classified  into  tvpes  according  to  the 
nature  of  the  double  pairs  and  of  the  conies  J\  i  and  A'o.  As  a  prc- 
linnnai'v  step  \\'e  shall  prove  the  theorem: 

V.  If  tin'  jiti/nf  /'  Uinl  tin'  liin'  p  fnnn  '/  ilnii!,!,'  jmir.  tfnn  i>  /.v  tin- 
pnliir  nf  r  with  rt'x/»'''f  tn  t/if  cmiii'  h  . 

To  prove  this  let  the  coordinates  of  /'  he  //,,  \\  here  i/t  is  t  IIP  solution 

of  (  7  )  for  p  —  p  ,  and  let  rt  he  the  coi'mli  nates  of  [>.    Then  r.  is  deter- 
mined frmii  (  1  )  \vlicn  .r,  is  replaced  by  //..    Then  from  (  1  )  and  (  7  )  \vc 


Pr, 


p  4-  —'•,  =  ( ",i 4-  ",,)  '/!  4- (",-.,+  «••)'/«+  ("..,+  ", ... ) >i... 

V  P^' 

These  last  equations  are  exactlv  those  which  determine  the  polar 
of  /'  with  respect  to  A',  and  the  theorem  is  proved. 

\\'e  now  proceed  to  the  classification. 

.(.  I. it  l\  lit-  it  n'mJciji'iirratfi  <'»nii:  l>v  a  proper  choice  of  coordi- 
nates its  equation  can  be  put  in  the  form 

so  that      flrii==  r/iio=  0,     ",r     ~",.'     ".,,=  -".,0     "]^;~".r 

If  tliert'  is  at  least  one  double  pair  of  which  the  point  is  not  on  the 
conic,  it  niav  be  taken  as  A  ( (I  :  0  :  1  )  without  chan^in^  t  he  form  of 
equation  (  1' ).  We  shall  then  have  a  =<!„.,=  Q,  The  correlation  is 
now  expressed  by  the  equations 


Neil  her  '/     n  or  nn   can  be  /.ero.    There  are  then  two  t  vpes  accord- 
ing as  ii     and  <t^    are  or  are   not    eipial: 

'1  \  !•]•:   I.  pn\  ---        it.  i'.,, 

P"',  =  "•'',- 

P"':  ~  •'  ,' 


02  TWO-DIMENSIONAL  (JKOMKTIIY 

The  conic-  A',  has  now  the  equation  J'~-{-'2  tt.r  ./•„  —  0,  and  the  correla- 
tion is  a  jiolarit \'  with  respect  to  this  conic,  ('onverselv,  any  polarity 
with  respect  to  a  nondegenerate  conic  can  be  expressed  in  this  form. 

The  equation  (  *  )  now  becomes  <t~{*[—pY=  0,  and  equations  (7) 
are  identically  satisfied  when  p  =  \.  Hence  in  <t  ]>nl<trit>/  ever;/  <•<>>•- 
rclnti'J  jH'int  dn<l  Uniform  /t  t/nn?,/<>  j>,tir.  The  eijuation  (_*i )  now 
becomes  <tu'^+  '2  v^t.,  —  0,  which  is  the  line  equation  of  A^.  Hence 
in  a  ]>"l<trity  the  conic*  J\i  anil  A',  coincide. 

TYPE   II.  pu[=         rrr.,, 


The  conic  A',  has  the  line  equation 

(<(  +  !')  'V'i!+  '"''";f  =  ^ 

or  the  point  equation 

4^rr/-,  +  (^  +  Mr;=0, 

and  the  relation  of  the  two  conies  A'  and  A"o  is  as  in  Fi^.  2").    Equa- 
tion (  * )  heconies         1  ,         , 

(I  -  p  )  (  n  -  lp  )  ( /.  —  dp  )  =  0, 

which   has  three   unequal   roots.    The  correlation   has  accordingly 
three  double  pairs:   namely,  the  point  „•/  and  the  line  !',(',  the  point 
I>  and  the  line  .I/.',  the  point 
(.'  and   the   line  AC. 

Tvpes  I  and  II  arise  from 
the  assumption  that  there  is 
a  double  pair  of  which  the 
point  lies  outside  the  conic. 
If  there  is  no  such  pair,  there 
must  he  at  least  one  of  which 
the  point  lies  on  the  conic. 
In  this  case  take  the  point  as 
/>'  (  0  :  I  :  o  )  without  changing 

the   form   of  equation  ('.').     My  theorem   V  the   line   of   the   double 

pair  which  contains  />'  is  the  tangent   />.(.     Then,  from  (1  ),  ti      -.-.(). 

\Ve  have  before  seen  that  ".,„ :     —  <'.,.,.  so  that   the  correlation  is  now 

p>/[=  .i.j-,  +  <in.r3. 

P"'.  ^        ",r'r 

u'=-  <*   *  +  •''• 


Fi...  '-'- 


LINEAR   TRANSFORMATIONS 


The  coefficient  al3  cannot  In1  /.ero  or  we  should  have  the  previous 
case.  The  equation  (S)  is  now  (n  —  pn  )  (a  —  p>i  n)  {J[  —  p~)=  0, 

and  the  solution  p  —  l  would  inve  a  point  not  on  A',  contrary  to 
hypothesis,  unless  n_^=(i^.  We  have,  finally,  for  the  equations  of 
the  correlation  : 

Tvn:  III.  pu[  =  '/./-.,  -f  /'./-3, 

K  =       "-'V 
P11'*  =  -  ;'-r,          4-  r,, 
where  a  =  k  is  not  excluded.    The  line  equation  of  A',,  is  now 


and  the  corresponding  point  equation  is 

I'-JT^  +  J--  4-2rtr:jra=0. 

The  two  conies  A"t  and  A'o  lie  therefore  in   the  position  of  V\^.  2»i. 

The  equation  (S)  for  p  has  the  triple  root  p—1^  and  the  cor- 
relation has  only  one  double  pair  consisting  of  the  line  point  />' 
and  the  line  A/1. 

/>'.  Lft  fhf  fnnii'  K  dc(fenernte 
info  tiro  intersect  iw)  utrn.iij'ht 
litifK.  We  niav  take  the  e<jua- 
tions  of  the  lines  in  the  form 


<tnn  =  0, 

a    :     —  n    ,      <i 

P.2  2,-i 

The  point    />'  is  a(_rain   taken 
as   the   point    of   u  double   pair 

and  is  therefore  transformed  into  a  line  through  /•'.  and  if  we 
take  that  line  as  .r]  =  0  we  have,  from  (  1  ).  </..,  -  0.  The  equation  (  s  ) 

is  nmv  <(i  +  p)8d-p)  =  rt. 

where  ii ^  cannot  be  y.ero  since  the  correlation  is  nonsinijiilar. 
The  root  p—  \  inves  the  point  /•'  as  a  point  of  a  double  pair. 
The  root  p~-\  j^ives  the  point  0 :  —  rt  .  :  <r  .,,  and  it  this  be  taken  as 
.  I  we  have  <i  .  —  0. 


04  TWO  DIMENSIONAL  (JKOMETKY 

We  have  then,  iiniilly. 

Tvi'K   I  V.  pn[  ---        <t.r}+  /-./-,, 


P"     -  •'•;,- 

where  the  equality  (|l  the  coefficients  is  not  excluded. 
The  conic  A'.,  has  now  the  equation 

nil':  +  f'~li~  =  (>, 

which  is  that  of  two  pencils  with  their  vertices  on  All.    The  relation 
of   A"    and    A",  is  shown   in   Fig.  1^7. 

'  '.  L>'t  fin'  I'l'iiir  A"  iliiji'th  rnli'  into  tiro 
i-i,i)n'i'i/>'nt  ttfrtrij/Jit  h'tii'x.  Take  the  equa- 
tion  of  A'  as  ..•_•  _  o 

•';!    —        • 

The  discussion  proceeds  as  in  the  pre- 
vious case  with  the  coefficient  n  placed 
equal  to  zero.  We  have,  accordingly, 

Tvi'K  V.     pir:  =        -  A./-.,, 


The  conic  A"_,  has  the  equation  iif  =  <>.  which  is  that  of  a  double 
pencil  of  lines  with  the  vertex  .1.  The  relation  of  the  two  conies 
A";  and  A'o  is  shown  in  Fig.  ^x.  The  equation  (S)  now  becomes' 

\R 

The  root   p  =  l    gives  the  point  .-I  as 

a  point  of  a  double  pair  of  which  the 
line  is    /!''.    The  root   p  -•  -1    gives 
anv  point  on  the  line  !'>< '.  so  that  it  M 
is  anv  point  on   IK'  it   is  a  point  of  a        y  t 
double  pair  the  line  of  which  is  AM. 

EXERCISES 

1.  Find  the  <i|uare  of  each  of  the  different  tvpes  of  correlations  and 
determine  the  tvpe  of  coll  i  neat  ion  to  which  it  belongs. 

2.  I'rove  that    it   /'  i--   a   point   on  A"t  the   tuo   tangents   drawn    from  /' 
to   A,   are    the    i  \\-o    lines    to    which    /'   corresponds    in    the    con-elation 
according  as  /'  is  considered  as  an  original  \«>\\\\  or  ;i  t  runs  formed  jioint. 


LINEAR   TRANSFORMATIONS  <).-) 

3.  Prove  that  if  //  is  a  tangent  to  A'.,  the  two  points  in  which  //  inter- 
sects A'    are   the   two  points   to   which  //  corresponds   in   the   correlation 
according  as  />  is  considered  as  an  original  line  or  a  transformed  line. 

4.  Take  any  point  /'.    Show  that  the  line  into  which  /'  is  transformed 
bv  a  correlat  ion  of  Types  II.  I  1 1,  V  is  a  line  which  connects  two  of  t  he 
four  points  of  intersection   with  A'    of  the  two  tangents  drawn    from/' 
to  A'.,.    Show  also  t  hat  t  he  line  which  is  transformed  into  /'  is  a  not  her  line 
connecting  the  same  four  points  of  intersection.     Determine  these  two 
lines  more  exactlv  and  explain  the  construction  in  Type  IV. 

5.  Take  any  line  ji.    Show  that  the  point  into  which  /MS  transformed 
by  a  correlation  of  Types  II.  I  I  I ,  V  is  one  of  the  four  points  of  inter- 
section  of   the   four   tangents  drawn   to  A'.,  from  the   points   in   which  // 
intersects  A"  .     Show  also  that  another  of  these  points  of  intersection  is 
the   point   which   is  transformed   into  />.     Determine  these  points  more 
exactlv  and  explain  the  construction  in  Tvpe  IV. 

6.  Show  that  if  every  point  lies   in  the  line   into  which  it    is  trans- 
formed by  a  correlation,  the  correlation  is  a  singular  one  of  the  form 


43.  Pairs  of  conies.  The  preceding  results  may  be  given  an 
interesting  application  in  studving  the  relation  of  two  conies  to 
each  other,  especially  with  reference  to  points  and  lines  which  are 
the  poles  and  polars  of  each  other  with  respect  to  both  the  conies. 

Let  y,/a..r,r,=  <>  (1) 

/     i       IA       I       A 


lie  two  conies  without  singular  points.  The  product  of  a  polaritv 
with  respect  to  (1)  and  a  polaritv  with  respect  to  ( '_' )  is  a  non- 
singular  colhneat  ion  which  mav  be  expressed  bv  the  equations 

P  (  ''\r>'\  +  f>iA  +  f>*A  >  =  f'irri  +  Wi  +  "V.T  (  :}  ) 

P  (  f>\;/\    +-  f> •••./-  +  ''.'.."'I!  )      :  "i:r''l      ^  ''-::•'".•  +  '<•..;•>'  • 

'I  he  fixed  points  of  the  collineation  (  •'">  >  are  identical  \\ith  the 
points  which  have  the  same  polars  with  respect  to  both  <  1  »  and  (  il  ). 
and  the  tixcd  lines  of  (  '•] )  are  ident  ieal  with  the  lines  which  have  t  he 


06  TWO    DIMKXSIONAL   (JKOMKTIIV 

same  poles  with  respect  to  (1  )  and  ('2).  Kach  fixed  point  of  (o) 
will  be  paired  with  some  fixed  line  of  (-\)  as  pole  and  polar.  These 
points  and  lines  we  shall  refer  to  briefly  as  common  polar  elements. 
We  shall  have  as  many  arrangements  of  common  polar  elements 
as  there  are  arrangements  of  fixed  points  of  ( •> )  and  may  classify 
them  into  the  types  ^iven  in  £  41. 

Tvi'K  I.  There  are  three  and  only  three  common  poles  A,  /•',  C 
(  Fi^.  '2'.' )  and  three  common  polars  .I/.'.  IK',  ('A.  To  pair  these  oft 
we  notice  first  that  no  point  can  be  the  pole  of  a  line  through  it. 

For  if   H  were  the  pole  of  />' 

.I/.',  for  example.  ('  would  be 
the  pole  of  either  AC  or  /!(', 
say  AC.  The  lines  AH  and 
.If 'would  be  tangent  to  each 
of  the  conies  (  1  )  and  ("2)  and 
A  would  be  the  pole  of  IK '. 
Then  if  I>  were  any  point 
whatever  on  lie.  and  E  its 
harmonic.1  conjugate  with  re- 
spect to  /,'  and  C.  the  line 
I-'. A  would  be  the  polar  of 
l>  with  respect  to  both  (1) 

and  (2).      Hence   the  conies  would  have   more  than  three   common 
polars.  and  the  eollineation   ( -\ )  would   not   be  of  Type  I.  £41. 

Therefore  the  triangle  is  a  self-polar  triangle  with  respect  to 
both  (1)  and  (2).  By  taking  this  triangle  as  the  coordinate  tri- 
angle, the  equations  of  the  conies  reduce  to  the  forms 

./•i2 +  ./•:+./•;=  0,  (4) 


P  •''.;  — 

\vhfi-t-.    by    £    H .    it  ^  -:    it     :    ,i  . 

Tin-  two  cidiics  (  1)  and  (  •"  )  intersect   in   four  distinct   points,  as 
is  easily  proved. 


LINKAK    TRANSFORMATIONS  HI 

Tvi'l".  II.  There  are  two  common  poles  ,1  and  C  (  I- "\\r.  •><• 
and  two  common  polars  AC  and  /!('.  The  point  C  must  he  tin 
pole  ot  one  of  the  lines  A*'  and  /!(' 
which  pass  through  it,  and  hence 
<'  lies  on  the  two  conies.  But,  (' 
cannot  he  the  pole  of  !'>(',  for,  if 
it  were,  .1  would  he  the  pole  of 
AC,  and  the  line  AC  would  he  taii- 
X'«'iit  to  the  conies  at  A  and  in- 
tersect ine;  them  a^ain  at  <\  wliieh 

is    impossihle.      Therefore     C    is    the       (  <_• 

pole    of    AC   and    J    of    lie.     If    we  F|(     .,() 

take   the   axes    of   eoi'irdinat es    as    in 
Type  I\",   sj    11,  the  equation   of  each   ot    the   conies   is   ot    the    form 

,v\r +'','•; +^vvv:<>.  (7) 

Without   cluing'illg'  the  position  ot    the  axes  we  mav  take  one  ot 
the  conies  as  t.-2  i    t.-2  ,    o  r  ;.  .     o  ,  ^  \ 

leaving   the   equjition    of   the   other   in    the    jjvneral    form    (^7).     'i  he 


p( 


P-'    -  '-   "•••'  -P 


P-'i  == 


That  this  should  he  of  TV pe  I  \  ,  >j  -1  1.  \ve  must  have  n ^  -••-  n  ,  n t   •    <i  . 

The  conies  (  1  )  and  (  '1 )  are  tangent  at  < '  and  intersect  in  t  wo  other 
points,   as   is   easdv   [irovi'd.     The  AI>, 

conii-s  ha\e  no  common  sell-polar 
I  naii'_;'le  since  t  here  are  in  .t  1 1  nve 
fixed  point  s  in  t  he  <•<  >1  lineal  ion  ( '.' ) . 

TVI'K  I  I  I.  There  is  a  line  H<  ' 
(  !•'!'_;•.  •'!  1  )  each  point  i  if  \\  hich  is 
a  common  pole  ami  another  com- 
mon pole  .1  not  on  liC.  The 
common  polars  consist  of  the  line  liC  and  all  lines  ilin>n<_:-Ii  I.  I; 
is  e\idelit  that  .1  ix  the  common  pole  of  /!<',  and  hence  /.'''  is  imt 


OS 


TWO   DIMENSIONAL  (JKOMKTHV 


tangent  to  tin1  conies.  'I'akc  as  // any  point  of  />v'and  take  ('  as  the 
pole  of  .!/>'.  'I'licii  . I /»'(' is  a  common  self-polar  triangle.  The  equa- 
tions <>f  t  lie  two  conies  may  now  he  written  as  in  Type  I,  ( 4  )  and  (•">), 
with  the  addition  that  now  <i}  -  </.,,  in  order  that  the  eollineation  (t!) 
should  he  of  1'vpe  II,  Jj  41.  Hence  the  equations  of  the  conies  are 

reduced  to  the  forms 

;/-,- 4- ./•;  +  .'•:=<>,  (10) 

j-{+j-*+atf=Q,  (11) 

and  the  eollineation  ('•])  becomes 


p.r,  =          rrra. 

The  two  conies  are  tangent  at  two  points,  namely  the  points  in 
which  the  line  IK'  meets  the  conies.  This  is  easily  seen  from  the 
equations.  \Ve  may  also  argue  that  if  IK'  meets  (!•')  in  A,  the 
point  /,  is  a  common  pole  of  the  line  A  I..  Hence  AL  is  tangent 
to  both  conies.  Similarly,  if  .!/  is  the  other  point  of  intersection 
of  IK'  and  (lo).  AM  is  a  common  tangent  to  the  conies. 

Tvi'K  IV.  There  is  one  common  pole  ('  (Fig.  •)-)  and  one  com- 
mon polar  /•''.  Hence  the  two  conies  are  tangent  to  IK'  at  (' 
and  tangent  at  no  other  point.  Take  any  point  on  the  conic  (1)  as 
.1.  and  the  tangent  to  (  1  )  at  A  as  A  /!.  J;\ 
The  equal  ion  of  (  1  )  then  is 
.!••+  f2  ./•/•„  =  0, 

while  that   of  (  -  ).  since   it    is  known 
only  to  be  tangent  to  IK'  at   < ',  is 

The  eollineation  (•>)  is  then  of  the  type 

P4  Vi+'Vsi 

p.r\        '/../'I  -f  .i..,:.  +  ,/../-... 

Iii    order    that     this    should    havo 
mdy   one    fixed    point    it    i-;   necessary 
and   snl'lieieiit    that    '/       </..    // .  ••-  ".     The   two   conies,   besides  being 
tangent    at     < '.    intersect    m    the    point    ./^  :./•,:./•,=  '/::  4  n..<i^.—  *  'if. 


LINHAi;    TRANSFORMATIONS  '.('.) 

If  this  point    is   taken   as  the  point  .1    in   the  coordinate   triangle, 
\\  e  have  >i    -    ".     The  equations  ot    the  conies  are   then 


and  the  culliueation  (  :'>  )  is 


p.  I'.,  - 


(I'M 


which   is  of  Tvpe  VI,    !j   11. 

As  noted,  the  two  conies  are  tangent   at    one  point  and  intersect 
in  am  it  her  point. 

TYI-K  V.  There  is  a  line  /!>'  (  l-'i^.  :}:} )  of  common  poles  and  a 
pencil,  with  vertex  ('  on  !'•<',  of  common  polars.  Kver\  point  on  ]'><' 
is  theretore  the  common  pole  ot  some  line  throiic'h  f ',  and  hence 
('  is  the  comnioii  pole  ot  IK'.  Hence  the  two  conies  are  tangent  to 
J1C  at  ( '.  \\'e  proceed  as  in  T\'pe  IV,  lnit  we 
iiou'  find  that  in  order  that  all  points  on  r , .--  0 
should  lie  tixeil  points  ol  the  i-ollineat  ion  we 
must  have  ti—(i,it^—ty.  The  equations  of  the 
ei  Uiics  tlit-refi  ire  reduce  to 


^  +  "A  +  -•'',•'';  ^  "^ 
and  tlu-  collineation  (  :>  )  liecomes 

PJ'(  =  -''i       +  "•'  - 
^  - 
= 


Tvi'K    \   I.     Fverv    point    of    the    plane    is    a    common    pole    \\iih 

I'opcct     to   the    t  \\  o    collies.      The    two    collies    are    iiliviolislv    identical. 

1  o  each  tvpe  ot  the  arrangements  tit  the  common  polar  elements 
corresponds  a  distinct  kind  ol  intersection  ol  the  t\\o  conies. 
(  'oiiverselv,  the  nature  ot  ihe  ciimmoii  polar  elements  ;>  detcr- 

ied   li\p  the  nature  of  the   inter.vcct  ions,   as   i>  easil\    pro\e<l. 


100  TWO-DIMENSIONAL  (IEOMETHY 

It  is  sometimes  important  to  find,  if  possible,  a  self-polar  triangle 
common  to  two  conies.  The  foregoing  discussion  leads  to  the 
following  theorem  : 

It  tit'n  rmi/i-s  iiiterxert  in  fiinr  distinct  points  they  have  one  and 
ail  /t  "in  i-'ii/i  ini'ii  si  If- f  i'i/ 1 1 /•  tridni/le.  If  tht'if  are  t(tinj<  nt  in  tiro  points 
///<//  luii't  tin  inliiu'ti'  number  of  rii)itiitnn  »elf -polar  triantjlex,  one  vertex 
(if  u'fiti'/t  is  /it  tin'  tiitiTtifi-tioit  of  tin'  i'ii/niiion  tan</ents.  In  <ill  other 
casts  f  t/'o  i/is///ii't  ci'H/i-s  //ii/'i-  no  I'otiinton  self- polar  triainjle, 

It  is  only  when  two  conies  have  a  common  self-polar  triangle 
that  their  equations  can  be  reduced  each  to  the  sum  of  squares 
as  in  Types  I  and  1 1 1. 

EXERCISES 

1.  Prove  that  the  diagonal  triangle  of  a  complete  quadrangle  whose 
vertices  are  on  a  conic,  or  of  a  complete  quadrilateral   whose  sides  are 
tangent    to  a    conic,   is   self-polar  with    respect    to  the   conic;   and,  con- 
versely, every  self-polar  triangle  is  the  diagonal  triangle  of  such  a  quad- 
rangle  and   such  a   quadrilateral.     Corresponding  to  a  given   self-polar 
triangle  one  vertex  or  one  side  of  such  a  quadrangle  or  such  a  quadrilat- 
eral may  be  chosen  arbitrarily.    Apply  this  theorem  to  determining  the 
common  self-polar  triangle  of  two  conies  in  the  position  of  Type  I. 

2.  Discuss  the  common  polar  elements  of  a  pair  of  conies  when  one 
of  them   has   singular    points,   obtaining   seven    types   corresponding  to 
the  seven  types  of  singular  eollineat  ions  given  in  MX.  1.  ?   11.    (Notice 
that  if  the  conic  i  1  )  consists  of  t  wo  intersect  ing  st  raight   lines,  the  point, 
of  intersection  /'is  the  singular  point  of  the  corresponding  eolliiieation, 
and  the  polar  //  of  /'  with  respect  to  the  conic  <  L' )  is  the  fixed  line.     If  the 
conic  (  1  )  consists  of  a  si  raight  line  taken  double,  t  hat  line  is  the  singular 
line  //,  and  its  pole  /'  with  respect  to  the  conic  <  L' )  is  the  tixed  point.) 

44.  The  projective  group.  As  we  have  seen,  the  product  of  two 
eollineat  ions  is  a  eollineat  ion,  and  the  product  of  two  correlations 
is  a  colliiieation.  It  is  not  difficult  to  show  that  the  product  of  a 
colliiieation  and  a  correlation  in  either  order  is  a  correlation.  The 
inverse  transformation  of  either  a  colliiieation  or  a  correlation 
always  exists  and  is  a  colliiieation  or  a  correlation  respectively. 
I  fence  we  have  t  he  t  he<  >rem  : 

Tin'    tntiilit/i   if  iionxi //i/nl<i  r   mill  iK'nt  i<m*   tiii'l    ii"iisi  ni/nlit  r  rnrt'i'/a- 

ft'i/IS     III      it      nlitn,'    Jurat     a     ijl'otlp,     i  if    H'liii'll      t//,'     cull  I  iK'ilt  li'iix   Jui'llt     (I 


LINEAR   TRANSFORM  ATK>NS  101 

Tliis  group  is  called  the  pruji'i-tirt'  <//'""/',  and  ///•o/Vr/'/'v  >/>•/>,  //cfr// 
consists  of  the  study  of  properties  which  are  invariant  under  this 
group. 

It  is  evident  then  that  project  ive  geometry  will  include  the  study 
of  straight-line  figures  with  reference  to  the  manner  in  which  lines 
intersect  in  points  or  points  lie  on  straight  lines.  Such  theorems 
have  been  illustrated  in  sj  :><>.  Lengths  of  lines  are  not  in  general 
invariant  under  the  projective  group,  and  projective  geometry  is 
not  therefore  concerned  with  the  metrical  properties  of  figures. 
The  cross  ratio  of  four  elements  is,  however,  an  invariant  of  the 
projective  group,  and  hence  the  cross  ratio  is  of  importance  in 
projective  geometry. 

IJv  means  of  a  collineation  anv  conic  without  singular  points 
mav  be  transformed  into  the  conic 


This  was  virtually  proved  in  ^  •>•">  when  we  showed  that  anv  equa- 
tion  of  the  second  order  with  discriminant  not  y.ero  maybe  reduced 
to  the  above  form.  l>ut  any  transformation  of  coordinates  is  ex- 
pressed by  a  linear  substitution  of  the  variables,  and  this  substitution 
mav  be  interpreted  as  a  collineation,  the  coordinate,  system  being 
unchanged.  Hence  anv  conic  without  singular  points  can  be  trans- 
formed into  any  other  conic  without  singular  points  by  a  collineation. 
Similarly,  any  conic  with  one  singular  point  may  be  transformed 
into  anv  other  conic  with  one  singular  point,  and  anv  conic  with 
an  infinite  number  of  singular  points  mav  be  transformed  into  anv 
other  which  also  has  an  infinite  number  of  singular  points.  Hence 
projective  geometry  rccogni/.es  only  three  types  of  conies  and  studies 
the  properties  which  are  common  to  all  conies  which  belong  to  each 
ot  the  types.  Such  properties  are  illustrated  in  the  theorems  of 
ij  •')!'.  where  the  distinction  between  ellipse,  hyperbola,  and  parabola 
is  not  made. 

In  projeetive  geometry  it  is  convenient  sometimes  to  consider  the 
properties  invariant  under  the  subgroup  of  collineat  ions.  The  corre- 
lations may  be  implicitly  employed  by  use  of  the  dualistie  property. 

45.  The  metrical  group.  \Ve  shall  proceed  to  study  the  eollinea- 
tions  which  leave  all  distance  invariant  or  multiply  all  distances 
by  the  same  constant  k.  For  that  purpose  it  is  convenient  to  use 


102  TWO-DIMENSIONAL  GEOMETRY 

(  'artesian  coordinates.    Sinn*  it  is  evident  that  all  points  at  infinity 
remain  at   infinity,  the  transformations  must  he  of  the  form 

p.r  =  n^r  +  <y/  -f  "./, 

*>.'/'=  V  +  'V/  +  V«  0) 

P?=t, 

or  in  nonhomogeneoiis  form 

jj  =  a  j-  +  r/.,//  -f  a  . 


Transformations  of  this  type  are  called  (iffine,  since  any  point 
in  the  Unite  part  of  the  plane  is  transformed  into  a  similar  point. 
\Ve  proceed  to  lind  the  conditions  under  which  an  alline  transfor- 
mation will  have  the  properties  required  above. 

If  (./-j,  //l  )  and  (.r.,,  //.,)  are  any  two  points  which  are  transformed 
respectively  into  (./•(,  i/()  and  (.<•!,,  if'.,),  then,  by  hypothesis, 


Since  this  must  be  true  for  all  values  of  the  variables,  we  have 


aia,+  1^=  0. 

From  this  follows  alu'ebraicallv  *''.,=  ±"1,  A^T'/.,.  Also  an 
an^le  can  al\\a\s  l»c  found  such  that  </  —  k  cos  (/>,  b  =  k  sin  0. 
Equations  (-)  can  then  be  written 

./•'=  /-  (./•  cos  cf)  —  if  sin  <£)  -|-  rt, 
//'  —  ±  £  (•''  sin  (/>  -f  //  cos  $)-\-li. 


s 


The  product  of  any  two  transformations  of  the  form  ( 
also  of  the  form  (  :>>  ).  This  can  be  shown  bv  direct  substitution, 
or  fnlldws  ^eometricallv,  since  (  -\  )  is  the  most  general  collincat  ion 
u  hieh  multiplies  distances  bv  a  constant.  It  is  also  cvidi-nt  that 


L1NKAK   TRANSFORM  AT  KLNS  ](}'.} 

the  inverse  transformation   of  (o)  exists  and   is  of  the  same  form. 
Hence   the    following   theoivm: 

I.  Tranxfurmatinnx  <>f  ttn'  form  (3)f<>r)/i  a  yruup  called  (Jn-  t/tt'trirnl 
i/rt'iiji  <f  fiilliiieuti'jnx. 

To  this  we  add  the  following  theorem: 

II.  Hi/  tli,'  nit-t  r'n-itl  ///'"(//'  '.-'   i-nUim-iltivnx  thf  riri-lf  [mint*  itt  infinity 
iii'i-  't'f/nr  fi.i',',1  nf  iittft'f/mni/i'il  icitli  i'ii<-h  of/tfr.     (  'vnt'crxt'/i/,  <it///  <•<>!- 

Inn  'it  fill     H'llli'h     IfiU'ix     tilt'     '•//•<•/,•     fiut/it*     ft.it  il     '//'     llttr/'t'/HtHi/CH     tJlflll 


This  follows  from  the  fact  that  minimum  lines  (  £  111)  must  he 
transformed  into  minimum  lines.  Since  the  line  at  infinity  is  fixed, 
the  points  where  the  minimum  lines  intersect  the  line  at  intinitv 
must  he  fixed  or  interchanged.  Theorem  II  may  therefore  In- 
n-stated as  follows: 

///.  Tlif  iiiftrli-nl  <//'">/}/  /IV//VX  inntriiint  tin'  curre  nf  xft-n/nl  fluxx 
i-nitxtxtin,/  if  tft,'  tn'"  circle  i>«ints  at  infinity. 

\Ve  shall  no\v  enumei'ate  certain  special  types  of  the  trans- 
f'  irmat  ion  (  '-\  ). 


T  {,,'=,/  + I,. 

This   is  of  Tvp»-   \',  vj  41,  the   line  of  fixed  points  hcing  the  line 
at    intinitv,  and    the   pencil   of   fixed    lines   heing   the  parallel    lines 

L  i  ri  i 

intei'sect  ing  in  </:/<:<>. 

The    translations    evidently    form    a    subgroup    of    the    metrical 
gi i MI  p. 

II.       Rntiltinn    ttlmllt    it    //./'.//   jin'lilt. 

It  the  lixed  point  is  the  origin,  we  have  the  transformation 


the 


104  TWO-DIMENSIONAL  (JKOMKTKV 

A  rotation  about  any  other  point  is  the  transt'onn  (  ^  ;">  )  of  If  by  '/'. 
Thus,  if  //'  is  a  rotation  about  (  </,  /<).  Ji'  —  TUT  \  whore  //'  is  the 

transfonnatioii     ,    ,  ,  ,      .      . 

(  .r  —  ti  =  (  ./•  —  a)  e<  >s  <p  —  (  _//  —  //  )  sin  cp, 

!  //'  —  li  =  (.r  —  it  )  sin  </>  +  (//  —  //  )  cos  (p. 

The  substitutions  Ji  and  li'  form  each  a  subgroup  of  the 
metrical  group. 

III.  ^Iininificutiun. 

•  s        I  I 

f  .c  =  A".r, 
M\     ,       , 

[  //  =  *!/• 

This  is  of  Type  II,  ^41,  the  ti\e<l  point  being  the  origin,  and 
the  line  of  tixed  points  being  the  line  at  infinity.  The  pencil  of 
iixed  lines  is  the  pencil  with  its  yertex  at  (  0,  0  ). 

A  magnification  .]/'  with  the  iixed  point  (  <i,  I)  is  the  transform 
of  M  by  7';  thus,  M'  —  TMT~  \  where  .)/'  is  the  transformation 

(.r'—  a  =  k(.r  —  «), 
•]I'{,f'-I.=  k  (//-/')• 

The  transformations  .17  and  M'  form  each  a  subgroup  of  the 
metrical  group. 

IV.  litjlfi-tinn  <>n  <i  tttrai'/lif  lint: 

If  the  straiht  line  is  the  axis  of  .r,  the  transformation  is 


This  is  of  Type  II.  ^41,  the  line  of  fixed  points  being  //  =  0. 
and  the  distinct  fixed  point  being  0:1:  0.  The  tixed  pencil  of  lines 
consists  of  the  parallel  lines  through  0  :  1  :  0. 

If,  now,  ('  is  a  transformation  of  the  metrical  group  (  :>),  it  is  not 
difficult  to  show  that  it  is  the  product  of  transformations  of  the 
types  we  haye  enumerated.  There  are,  in  fact,  two  main  divisions 
of  the  metrical  transformations,  namely, 

CLASS  I.    Mi-trii'iil  Irnnxfnrmatinnx  nut  /'//rn/ri//</  </  /v  //<-.-/  /•<//. 
Consider  f        TMli.    It  is  evident  that  I'  is  given  by  the  equations 

'./•'  =  /•(./•  cos  (f)       //  sin  (f)  >  -f-  <i. 


and   that,   crmvcrsely,  any  transformation    of   this   type   can    be   ex- 
pressed   as    the    product     '/'.)//,'. 


LINEAR   TRANSFORMATIONS 

CLASS  II.   Metrical  trunxfurinatiuns  involving  u  r*'flei-ti<,/i. 
Consider  L'n  =  TSMli.    It  is  evident  that    ('  is  of  the  type 

(  ./•'  =  /c(.r  cos  (/>  —  y  sin  <£  )  -f-  '/, 
-  [  //'  =  —  /c  (  jc  sin  $  +  _y  cos  <j>)  +  /», 

which  can  also  be  written 


fj-'  =  /.•(./•  cos  <£  4-  //  sin  (/>)  -f-  </, 
-       '  =  /•  (  .r  sin 


If  COS  0  )  -f-  l> 

by  re-placing  0  by  —  (/>,  an  allowable  change,  since  0  is  any  angle. 

Conversely,  any  transformation  of  type  ('.,  can  be  expressed  as 
the  product  TSMlt. 

'1  he  transformations  ('  form  a  subgroup  of  the  metrical  group. 
The  transformations  I ".,,  however,  do  not  form  a  group,  since  the 
product  of  two  such  transformations  is  one  of  the  form  U . 

46.  Angle  and  the  circle  points  at  infinity.  I>y  the  metrical  group 
angles  are  left  unchanged.  This  is  evident  from  the  fact  that  any 
triangle  is  transformed  into  a  similar  triangle.  Also  the  cross  ratio 
of  anv  two  lines  and  the  minimum  lines  through  their  point  of  inter- 
section is  e([ual  to  tlu-  cross  ratio  of  the  transformed  lines  and  the 
minimum  lines  through  the  transformed  point  of  intersection,  since 
minimum  lines  are  transformed  into  minimum  lines.  This  suggests 
a  connection  between  this  cross  ratio  and  the  angle  between  the 
two  lines.  We  shall  proceed  to  lind  this  connection. 

Let  the  two  lines  be  /  with  line  coordinates  ?v,  and  /,  with  line 
coordinates  n\.  The  coordinates  of  any  line  through  the  point  ot 
intersection  of  /  and  /  are  uf=  '',4-X'/',.,  and  this  is  a  minimum  line 
when  a t  satisfies  the  line  equation  of  the  circle  points  at  inlinit\, 
namelv, 

This  gives  for  X  the  equation 

where  .-1  =  MY  +  M'.r,        /•' =  l'\<\  +  >'.,/r.,.        ('—rf+VJ. 

Lei    us    place  X,  : 

.1 

-  /;      /\Tr-/;J 

A     — 


Kit',  TWO-DIMENSIONAL  GEOMETRY 

and   call    >HI   the    niininiuin    line   corresponding   to   X  ,   and    ;//.,   the 
minimum  line  corresponding  to  X,.    Then  (^  1-5) 

X.       -  /;  +  /\  AC—  //• 

(/  /  ,  „>>,<  )=  —  =  - 

\    -jt-rtAc-jr- 

Now  the  point  eqmttions  of  /,  and  /.,  are  respectively 

<V'  +  '',.'/  +  >:J  =  0, 
wy  +  u\,y  +  trat  =  U, 

and  if  (/>  is  the  angle  between  them, 

>\"\+  t'.-ir.,  11 

cos  $-—  = 

\  l'+  r~\  tr-  +  tc:       \  AC' 


_ 

sin  9  =  _  -  • 

VAC 

,P,        ,  Xj       -  cose/)  ±  /sin  <^> 

1  herefore 

X.,       —  cos  9  ^  t  sin  9 


i         X 
whence  9  =  ±     log  -    • 

L       "    \ 

The  ambiguity  of  sign  is  natural,  since  an   interchange  of  \i  and 
X    would  change  the  sign  of  <b.     We  have,  therefore, 

•2  o  O 

9  =  ±  0  log  (//.,,  injii.,). 
Tin-    K/ii/li-    In'tll't't'H     (!/'<>     li/ti'X    ix    therefor?    r<ju<ll    t"  tinli-tt    tin' 


If    9-     ^  -       L=   -1,    and,    converselv,    if       1=  — 1,    9=    '  +  k-rr. 
•>       "x  ^  •> 

.  .  _         A ,  A., 

I  lelice 

/''/•/"  »'/>'•///:!/•  linfx  until  Ac  tiffined  <ix  Innx  H'hirJi  <tr<'  lutrtiUiHlt' 
i-niij iii/iit,  x  irit/t  ri-xjn'1-t  t/i  (/!,•  minimum  Itnrx  t/tr»u<//i  their  [mint  <>f 
1/ttiTxi'ctiv/i, 


CHAI'TKR    VII 

PROJECTIVE  MEASUREMENT 

47.  General  principles.  The  results  of  the  last  section  surest  a 
generalization,  to  be  made  liv  replacing  tin-  circle  points  at  intinitv 
hv  the  general  curve  of  the  second  class, 

V./a  >/,",-<>,        Utl  =  .•!„.)  (1) 

which  we  sluill  eall  the  fundamental  <•/////,•.  Let  /  and  /,  (  Fig.  o4  ) 
be  anv  two  lines,  and  let  t  and  ta  be  the  two  tangents  which  can  be 
drawn  to  the  fundamental  conic  from  the 
point  of  intersection  of  /(  and  /,.  Then  the 
nrojective  an^le  between  /,  and  I  is  defined 

I  J  CT  1  '2 

by  the  equation 

4_  (//,)  =  M  log  (//,,,  //,,),  (-1) 

where    .17    is    a   constant    to   be    determined 
more   exactly  later. 

This  satisfies  the  fundamental  require- 
ments for  the  measurement  of  an  angle, 
since  it  attaches  to  every  angle  a  definite 
numerical  measure  such  that  the  sum  of  the  measures  of  the  parts 
of  a  whole  is  equal  to  the  measure  of  the  whole.  To  prove,  tin- 
latter  statement,  notice  that 


Now.  if  /^  /,.  and  /,  are  three  lines  of  the  same  pencil,  with  coor- 
dinates \^  A,,  A.,  respectively,  and  the  coi'ml  mates  of  the  lines  /( 
and  /.,  of  the  same  pencil  are  taken  as  0  and  s_,  we  have 


Henc 


4  ( 
107 


=  4  ( 


108 


TWO-DIMENSIONAL  (IKo.MKTKV 


Dualistieally,  if  the  fuiulamental  conic  does  not  reduce  to  two 
points  its  equation  can  IK-  expressed  in  point  coordinates  as 


Then,  if  I\  and  /_!  (  Fig.  of>)  are  two  points,  and  7',  and  7',  are 
the  two  points  in  which  the  line  /,'/.!  cuts  the  conic,  the  projective 
distance  /,'/.!  is  delined  by  the  equation 

dist.  (  /;/;  )  =  A'  log-  (  l[If   7\  T..),   (4) 

where  A'  is  to  be  determined  later.  It  is 
shown,  as  in  the  case  of  angles,  that 
dist.  (/;/:)  4-  dist.  (/,:/;)  =  dist.  (A'/')- 

The  analytic  expression  for  distance 
and  angle  in  terms  of  the  coordinates  of 
the  points  and  lines,  respectively,  may  F](.  ,,- 

readily   be    found.     Take,    for   example, 

equation  (4).  If  //.  are  the  coordinates  of  /,'.  and  z{  the  coordinates 
of  ./!.  the  coordinates  of  7\  and  71,  arc  //(.—  A  .?.  and//,—  A2?,,  where 
A,  and  A0  are  the  roots  of  the  quadratic  equation 


which  we  write  for  convenience  in  the  form 


We  will  take 


ro,  ,  4- 


—  co(i)_ 


and 


A  = 


Then,  1)N-  the  definition  ( i2 )  and  theorem  III,  ij  lo,  we  have 


But 

\  ">„•     -    \    to^  —    tovi/to  a)!/./")z: 

and  therefore  we  have,  as  the  final  f<>nn, 

i-  to,, 4-\  to;  —  to    to  , 

dist.  (///.)=  -2  1\  log    •- 


PK<  ).I  KCTIVK   MEASUREMENT  1(1'.) 

There  is  of  course  free  choice  as  to  which  of  the  two  values  of 
X  is  taken  as  X  .  To  interchange  \l  and  X,,  is  simply  to  change  the 
positive  direction  on  the  line. 

The  distance  between  two  points  is  zero  when  the  two  points 
are  coincident  or  when  the  line  connecting  them  is  tangent  to  the 
fundamental  conic,  since  in  the  latter  case  X;  =  X,.  The  tangents  to 
the  conic  are  therefore  analogous  in  the  protective  measurement 
to  the  minimum  lines  in  ordinary  measurement. 

The  distance  between  two  points  is  intinite  when  X(  or  X,  is 
zero  or  infinity.  This  happens  only  when  /,'  or  I'  is  on  the  funda- 
mental conic.  That  is,  y^//;/1*  on  thf  fundainrntal  n,  ///'<•  art'  at  mi 
infinite  dittanfe  from  all  other  points. 

Similarly,  consider  equation  ('!).  If  '-.and  >/•,  are  the  coordinates 
of  /,  and  /,  respectively,  the  coordinates  of  ^  and  ta  are  \\—  \l"\  and 
!•-—  \0wv,  where  \l  and  X.,  are  the  roots  of  the  equation 


which  mav  be  written 

n      -2xn,,+  \2n.=  o. 


r         i  ^      -.,,.    N     , 

t  we  take  \^  =  -  — 


we  have,  bv  (-),  _ 

\  O     +  x  p-   ._  q    r> 

4  (,-;/•  )  =  .l/log-1  -iM/log  .....  .        (  ,  > 

1  \  \  ",,.n,,,,, 

An  angle  is  zero  if  /;  and  /,  coincide  or  if  ^  and  /,  intersect  on  the 
funilamental  conic,  for  in  the  latter  case  X^X.,.   That  is.  •///////.  .v  '/•///••// 

llit>  1'Xi-i-t  'it   il/i    nit'fllit'    <l  ixtil  in'*    null:'    'I  Zi'T'i  >•'/>'//'     il'ltji   .  it,  -/i    nf/n  r.        I   hc\' 

are  therefore  analogous  to  parallel  lines  in  Euclidean  measurement. 

The  air_;'le  between  two  lines  is  intinite  if  either  line  is  tangent 
tn  the  fundamental  conic. 

Fri'in  the  definitions  we  have  the   following  theorem: 

/*/•';/>'<•//'•<      il'txtttllfi'    il/nl    <//"//,     ,ir>-    il/l<-]i<l//<l'>]    I"/    t)i>     '/f">IJ>    ';'    1'iJftfl- 
t-,ltf'i/ifs    ll-fiii'Jl    /'it/',     f/i,'    t'n  inLl  Hi'  Kf<tl    i-i'/ii''   i  ii  '''I  I'l'l  nt  . 

\Ve   shall    now    pmceed    to    discuss    imu'e    in    detail    three    cases, 
according   ti>   the   nature  of   the   fundamental   conic. 


110 


T\V(  >-])!  M  KNSK  )N  AL  (J  K(  >M  KTH  V 


48.  The  hyperbolic  case.  We  assume  that  the  fundamental  conic 
is  real.  It  may  then  be  brought  by  proper  choice  of  coordinate 
axes  to  the  form 

in  point  coordinates  and  to  the  form 

in  line   coordinates. 

The    conic    divides   the    plane    into   two   portions,    one    of 
we  call  the  inxiilc  of  the  conic  and  which  is  characterized 
fact  that  the  tangents  to  the  curve,  from 
any  point   of  the   region    are   imaginary. 
The   <Dit*t'i/<'   of    the    conic   is   the   region 
characterized  by  the  fact  that  from  every 
point    of    it    two    real    tangents    can    be 
drawn.     We  shall  consider  the   inside  of 
the  conic. 

If  /l  and  /,  (Fig.  ^h")  are  two  real 
lines  intersecting  in  a  point  inside  the 
conic,  X  and  Xo  of  equation  (7),  §47, 
are  conjugate  imaginary.  Let  us  place 
Xj  =  >v"/>,  where 


« 

cos  9  —  — 


.    ,     v/n  n    -n* 

s  9  =  -      _       — ,  sin  (p  = 

Then  Xa  =  rc~{*  and 

Since    it   is   desirable   that   the   angles   which   a    line    makes   with 
another  should   differ  by  multiples  of  TT,  we   shall   place   M 

and    have,  as  the    complete   definition    of    the    angle  0   between    the 
Iill('s  ;,  ;U1(1   ^  0  =  <£  +  mr: 

whence  cos  0  =  ±      /  •  ( :>> ) 

7T 

Two  lines  are  perpendicular  to  each  other  when  0  —  (  '1  n  -f  1  ) 

For  that  it  is  necessarv  and  snflicient  that  -1.     The  two  lines 

X 


I'KO.IKCTIVK   MKASl'KKMKNT  111 

arc  then  harmonic  conjugates  with  respect  to  t{  and  t^.  This  has  a 
H'eoiiictric  meaning,  as  follows:  Let  /'  (  I''ii^.  •>'»)  lie  the  point  ot  in- 
tersection of  /t  and  /„,  [>  the  polar  of  /',  I.  and  /,„  the  intersections 
of  i>  with  /  and  /,  respect  ivelv,  and  7'(  and  7',  the  intersections  of 
the  conic  with  f  and  f:  respect  ivelv.  7'.  7',,  ^,  f.,,  hein^  imaginary, 
are  not  shown  in  the  figure.  Then  l>v  VI,  ^  -54,  7'  and  7',  lie  on  y>, 
and  by  I,  ^li'»,  (1^1..,.  7\  7',  )  (  /y.,,  M,).  Hence,  in  order  that  the 
two  lines  /  and  /.,  should  he  perpendicular  it  is  ueeessarv  and  sufli- 
cient  tliat  I,  and  L.,  should  he  harmonic  conjugates  to  7'  and  7',, 
and  hence  (VIII,  vj  ->4  )  /.  must  lie  on  the  polar  of  /.  ,,  and  /,  , 
must  lie  on  the  polar  of  /,  .  P>ut  the  polars  of  I.  and  L,  jiass 
through  /'  liv  \',  Jj  ^4,  and  therefore  /  is  the  polar  of  /.,,  and  /, 
is  the  polar  of  /,  .  Hence  f«r  lirn  Ihn'x  /•/  /<»•  pt<rpt>n<H<'l(ltir  it  i* 
th't't'ttxiiri/  iiH'l  xiillii-ii-nt  f/mf  >''/'•//  xh<>ul<l  jmxx  thr»n>j}i  (/n*  )><>1<'  "f 
thf  nt  JUT. 

(  'onsider  now  the  distance  het  ween  two  points  /,'  and  /.;  (  I-'i^'.  oti  ) 
inside  the  conic.  Then  X  and  X,  of  (•">),  ^  17.  ai'e  hoth  real,  and 
hence  if  the  distance  /[/'  is  to  he  real  we  must  take  A"  as  a  real 

ipiantitv.     Let   us  place    A"  =      ••   where  /."  is  real.     \\'e  have,   for  the 

distance, 

"          X.        ,  .       w    -f-  \  or..  —  <i>.  ,(:) 

—  .  (  4  ) 


If  \ve  \\'i'ite  </  for  dist.  (  //,  ^',  )  we  have,  from  (4), 


\\'e  have  already  noted  that  it  /,'  is  inside  the  conic  and  /._,'  on 
the  conic,  the  distance  /J/.;  bt'eoines  infinite.  If  /'  is  inside  the  conic 
and  /.,'  outside  of  it,  X,  and  X.  in  eijiiation  f  1  )  have  opposite  Minis. 


112 


TWO-PIMKXSIOXAL  (JEOMETKY 


ami  the  distance  1(1',  becomes  imaginary.  J  f ,  thru,  we  can  imagine 
a  being  living  inside  tin-  conic  and  measuring  distance  and  angle  by 
the  formulas  ( ."> )  and  (J-\  ),  the  (.-nine  would  lie  for  him  at  an  infinite 
distance,  and  the  region  outside  wonld  be  simply  nonexistent,  a 
mere  analytic  conception  in  which  a  point  means  simply  a  pair 
of  coordinate  yalnes.  Such  a  being  would  haye  a  nnn-Euclidean 
</*'"//n  ?/•*/  ot  the  type  named  Lobachevskiim. 

We  haye,  of  course,  based  all  onr  discussion  on  the  assumption 
of  the  Euclidean  axioms,  and  the  inside  of  onr  fundamental  conic  is 
simply  a  portion  of  the  Euclidean  plane.  It  lies  outside  the  scope 
of  this  book  to  show  that  by  a  choice  of  axioms,  differing  from 
those  of  Euclid  only  in  the  parallel  axiom,  it  is  possible  to  arrive 
at  a  geometry  which  for  the  entire  plane  has  properties  which  are 
exactly  those  of  the  interior  of  our  fundamental  conic,  with  the 
protective  measurement  here  defined.  Such  a  discussion  may  be 
found  in  treatises  on  non-Euclidean  geometry.  The  inside  of  the 
fundamental  conic  is  a  picture  in  the  Euclidean  plane  of  the  non- 
Euclidean  geometry.  We  shall  proceed  to  notice  some  of  the  most 
striking  properties. 

We  first  notice  that  if  /.  A"  (  Fig.  -\7 )  is  a  straight  line  and  /' 
a  point  n«t  on  it.  there  go  through  /'  t\vo  kinds  of  lines,  those  which 
intersect  /.K  and  those  which  do  not. 
The  latt'T  lines  are  those  which  in  the 
entire  plane  intersect  L K  in  points 
outside  the  conic,  but  from  the  stand- 
point of  the  interior  of  the  conic  thev 
\\\\\>\  be  considered  as  not  intersect- 
ing LK.  The  two  classes  of  lilies,  the 

mt  er>eet  inir    and    the    lion  intersect  in<r. 

Fn,.  :: 

are  separated    from   each  other  by  two 

lines   /'/.   and    /'A",   which    intersect    L  K  on    the   conic: 


infinity.  Thoe  lines  we  call  /»//•<///</  lines,  and  sav  that  tftri'n/t/h  a 
jaunt  nut  <i/i  ,i  xf/;i/i//if  //'//,•  f, in  },,'  ilriin-n  f/i'n  ////rx  ?»//v///»7  t<>  tlmt 
*f, •<"',//,/  tint: 

The  aii^le  which  a  line  parallel  to  1. 1\  through  /'  makes  with 
the  perpendicular  to  /,  A"  is  called  the  ////_<//.•  nf  pnraU<'lixii>*  and  is  a 
function  of  the  length  of  the  perpendicular.  To  compnte  it.  let 
us  take  U\  as  jr  —  0.  the  point  /'  as  //_,  and  the  equation  of 


PKO.JKCTIYK    MKASrKK.MKNT  Ho 

tlie  conic  sis  ./y  +  .r:  —  j\:  —  0.  The  pole  of  LK  is  (  1  :  0  ;  0  ).  The 
line  J'/i  is  perpendicular  to  LK  when  it  passes  through  the  pole 
of  LK.  Its  equation  is  therefore  //..-''.,—  //.,•''.--  ",  and  it  interM-cts 
LK  in  7,'  (0  :  //>:  //;i). 

Hence,  if  p  is  the  length  of   /'//  we  liave,  from  (•">), 

,  /'      V/^T-.'/J1  •   ,  /'  .'/i 

cosh  -  =  —  i  smh      =  -  ( »i  ) 

The  point  K  is  the  point  (0:1:1),  and  the  equation  of  I'K  is 
(//.,  —  //.,).  '"j—  //j-''., -(-  /^-''.j--  ''•  Hence  to  lind  the  un^'le  between  /'A" 
and  /'A'  we  have  to  place  in  (oj 


"'l=  !'.,—  .'/;,'          "'3=   -  //p          "'3= 

There  results,  Avith  the  nid  of  ('»), 


It   appears,  then,  tliat   the  single  ^  is  a  function  of  p.     We   sha 
place,  following  Lobaelievsky's  notation, 


Our  last  equation  then  leads  with  little  work  to  the  final  result  : 
tan  JTT  (»=,•-';•  (7) 

This  result  is  independent  of  the  fact  that  it  has  been  obtained 
for  the  special  line  .r  =0  and  the  special  form  of  the  equation  ot 
the  conic  since  no  t  ran  s  tor  mat  ion  of  coordinates  alt  ers  t  he  project  i\  c 
angles  or  distances. 

It  in  formula  (••>)  \ve  consider  //.  as  a  lived  point  f  and  replace 
zi  bv  a  variable  point  ./',„  at  the  same  time  holding  the  distance  </ 
constant,  we  ha\c 


as  the  equation  of  the  locus  ot  a  point  at  a  constant  distance 
from  a  fixed  point.  This  locus  is  called  a  pseiido  circle.  From 
tin1  form  of  (S)  it  is  obvious  that  the  pseiido  circle  is  tangent  to 


114 


TWO    DIMKNS10NAL   CKOMKTRY 


tlie  fundamental  conic  ror/.  =  0  at  the  points  in  which  the  latter  is 
cut  1)V  the  polar  &>,,,.  =  0  of  the  point  //,.    There  are  three  cases: 

I.  The   point   ('    lies    inside    the    conic    (Fig.  -5S ).     The    pseudo 
circles  with  the  center  //,  are  then  closed  curves  intersecting  the 
conic  in  imaginary  points. 

II.  The  point   ('  lies  on  the  conic  (Fig.  :W),  and  the  distance  of 
each  point  from  //,  is  infinite.    The  pseudo  circles  are  tangent  to  the 


Fit;.  08 


Fit;.  3!1 


conic.  They  art-  the  limiting  cases  of  the  pseudo  circles  of  Case  I 
when  the  center  recedes  to  infinity  and  the  radius  becomes  infinite, 
and  are  called  in  non-Euclidean  geometry  limit  circles  or  horicvcles. 
III.  The  point  C  is  outside  the  conic  (  Fig.  40),  and  the  radius 
is  imaginary  so  that  points  of  (S)  ljc>  inside  the  conic.  The  straight 
line  (!>„,=  "  is  one  of  these  pseudo  circles,  and  the  others  are  the 
loci  of  points  equidistant 
from  this  line.  To  prove 
the  latter  statement  draw 
anv  straight  line  through  ('. 
It  intersects  the  polar  of  < ' 
at  1!  and  the  pseudo  circle 
in  two  points  one  of  which 
is  (,>.  Then  ('/,'  and  <'(,)  are 
constant,  and  hence  Ii(t>  is 
constant.  In  t  his  ^cornet  rv. 

then,  the  locus  of  points  equally  distant  from  a  straight  line  is 
not  a  straight  line,  but  a  pseudo  circle  with  imaginary  center  and 
imaginary  radius.  It  is  called  a  hypocycle. 


PROJECTl  VK    M  KASU  K  KM  KNT  1  1  :, 

EXERCISES 

1.  Consider  angle  and  distance  for  points  outside  the   fundaiuenUil 
conic,  especially  with  reference  to  real  and  imaginary  values. 

2.  (/(instruct  a  triangle  all  of  whose  angles  are  /.ero. 

3.  Compute  the  angle  between   t  \Vo  lilies  of  /ero  length  and    Ix'tweeli 
anv  line  and  a  line  of  /.ero  length. 

4.  Prove  that   the  sum  of  the  angles  of  a  triangle  is   less  than  two 
right  angles. 

49.  The  elliptic  case.  We  assume  that  tin-  fundamental  conic  is 
imaginary.  It  mav  lie  reduced  l>v  proper  choice  of  coordinates  to 

tlir  t'"nn  w,,.=  ./v4-.r;  +  .r;  =  0  (1  ) 

in  point  coi'tnlinates  and  to  the.  form 

12KH=z/12  +  ?/|  +  ?/*  =  0  (-2) 

in  line  coordinates. 

Since  the  tangents  from  anv  point  to  the  fundamental  conic 
are  imaginary,  the  problem  of  determination  of  angle  is  the  same 
here  us  in  the  hyperbolic  case,  and  we  have 


Any  straight  line  connecting'  the  two  points  I\  and  /.!  meets  the 
conic  in  imaginary  [mints,  and  if  I[  and  /'  arc  real  points,  the 
quantities  \1  and  A.,  in  (•>),  £  47,  are  conjugate  imaginary.  Hence, 
it  the  distance  between  two  real  points  is  to  be  real,  we  must  take 

K  as  pure  imaginary.     We  will   place    l\  •  where  k  is  real. 

Placing  X(  =  /•«•"•',  where 

(Oi  .  .  \    ft)      O)      —  (i)' 

(•(  is  (p  =  .  Sill  (/)  -  •  "  < 

\    (i)      M  \    f<)      (I) 

and  representing  the  distance  (  y,.~,  )  l>v  '/.   we  ma\'  n-duce   formula 
(  •>  ),   vj  47,  to  the  form  ,/  (l) 


'I  wo  real  points  arc  always  at  a  I'mite  distance  from  each  other, 
since,  as  shown  in  £47,  an  infinite  distance  only  results  when  one 
ot  the  points  is  on  the  fundamental  conic. 

( 'oiir-idcr  the  change  in  <l  a^  r,  moves  along  a  straight  line.  // 
being  fixed.  In  the  he'nmiiii"'  of  the  motion,  when  ;  coincides 


116 

\vith  // ,  c 


TWO-DIMENSION  .VL  (}  EOM  ETK V 

:  •.  and  the  sign  of  the  radical  must  be  taken 
\  (t),..,(0, ,, 


that  cos      =  1   and  it  —  0.     As  zi  moves  away  from  >/i  the  signs 


of  the  (plant ities   on    the   right-hand   side  of  equation   (4)  remain 
positive  and  </  increases  until  zt  reaches  a  point  on  the  line  (^UJC=  0, 

/ 


(Fig.  41  ).  the  polar  of  //,.   Then 

,/  .„- 

cos      =  n  and  it  /.•.    This  is 


true  of  all  lines  through  //. 
and  for  either  direetion  on  any 
such  line.  Hence  the  straight 
line  fD,/,=  0,  which,  by  §  4<S, 
is  perpendicular  to  all  lines 
through  //,,  is  at  a  constant 
-k 


distant1 


from    //(    in    all 


ii 


directions. 

Consequently,  if  we  start  from  //,  and  traverse  a  distance  -rrk  on 
any  line  through  //.  and  in  either  direction,  we  return  to  yt.  There 
are  two  eases  of  importance  to  be  distinguished: 

CASK  I.  All  straight  lines  may  be  considered  of  length  Trk. 
The  coordinates  //.  always  refer,  then,  to  a  single  point.  All  straight 
lines  intersect  in  one  and  only  one  point,  then'  are  no  parallel 
lines,  and  two  lines  always  bound  a  portion  of  the  plane.  This  is 
the  Kit'mannian  </<'<>iiirtri/.  It  may  be  visuali/ed  by  drawing  straight 
lines  from  a  point  outside  the  plane  and  considering  each  point  of 
the  plane  as  represented  by  one  and  only  one  ot  these  lines. 

('ASK  II.  All  straight  lines  may  be  considered  of  length  '2  TT/C. 
When  we  traverse  the  distance  TT/,'  on  a  line  from  iff  and  return  to 
//,,  we  shall  consider  that  we  are  on  the  opposite  side  of  the  plane 
and  need  to  repeat  the  journey  to  return  to  our  starting  point. 
Any  coordinates  // ,  then,  are  tin.;  coordinates  of  two  points  lying 
on  opposite  sides  ot  the  plane.  Two  straight  lines  intersect  in  two 
points,  there  are  no  parallel  lines,  and  two  lines  inclose  two  por- 
tions ot  tin-  plane.  We  call  t  his  x/Jn'ri'-iil  i/coi/irfr//,  since  it  inexactly 
that  on  the  surface  of  a  sphere.  It  is  also  the  geometry  of  the  hall- 
lnies  or  rays  drawn  to  the  plane  from  a  point  outside  of  it. 


PRO,JE<.TIVE  MEASUREMENT  117 

EXERCISES 

1.  Construct  a  triangle  all  of   whose  angles  are  ri^ht  angles. 

2.  Prove   that    the   sum   of   the  angles   of  a   triangle    is   greater  than 
two  n^ht  angles. 

50.  The  parabolic  case.  We  may  consider  that  the  fundamental 
conic  is  one  which  contains  singular  points  or  singular  lines. 
There  are,  then,  the  two  possibilities  of  the  point  equation  repre- 
senting two  straight  lines  or  oi  the  line  equation  representing 
two  points.  The  former  possibility  has  little  interest,  and  we  shall 
consider  only  the  case  in  which  the  line  equation  represents  two 
points.  There  are  two  cases  to  distinguish: 

CASK  I.  Tht1  ("'»  p»!nts  <ir>'  i//i<i</iit<irt//.  We  may  take  them  as 
the  two  points  1:±/:0,  and  the  line  equation  oi!  the  fundamental 
conic  is  then  O  __  -j  ,  ,,-___()  i 

The   formula   for   an^le    nuiv  be   modiiied    as    in    ^  4*,    \\ith    the 

O  v  i1 

result   that 


'1  he  point  equation  oi  the  fundamental  conic  does  not  exist  and 
the  distance  formula  ('!),  sj  47.  cannot  be  immediatelv  applied. 
\\  e  ma\'  proceed,  however,  1>\'  a  method  oi  limits.  Jn  place  ot  (1  ) 

Wl'wi11  xvrite  3s  ' 


and   from  this  we  iind,  as  in  £  4S, 
sinh  'f       /  N  €('/^~  •//-;-'1  }~_^:-e( 


as  t~  ==  i)  and  h  —  s.    in  >iich  a   manner  t  hat    l.im  //,  \  t       1 .     We  ha\  e 


118  TWO-DIM  KNSIONAL  GEOMETRY 

If  we  take  ./'.,  —  0  as  the  line  at  infinity,  the  points  1:±/:0 
become  the  circle  points,  and  the  formula  (l!)  for  angle  and  ( ;"> ) 
for  distance  become  the  usual  Cartesian  formulas.  The  <reometrv 

O  * 

is   Kuclidean.     We  have  this  result  : 

Km'UJi'iin  itii'iixut'ciiii'iit  ix  tt  speciul  m.sv  <>f  ///v//<-<Y//r  /nc<<xi<r<'i/t<'>tf. 

CASK  II.  The  funilunii'tilitl  puintx  «/'<•  /•>'<(/.  We  may  take  them 
as  l:i;  1;(>.  The  line  equation  of  the  fundamental  conic  is  then 

i  luu  =uf—u::=  0.  (ti) 

Since  through  every  real  point  there  go  two  lines  of  the  pencils 
detined  bv  ('>),  it  is  necessary  to  take  the  constant  K  of  §47  as 
real  if  real  lines  are  to  make  real  angles  with  each  other.  We 
will  take  l\  —  \  and  lind,  by  a  discussion  analogous  to  that  used 
in  vj  4S  for  finding  </, 

cosh^  (7) 

\  /Y  —  r.r  v  M'j"  —  /r? 

The  formula  for  distance  mav  be   found  as   in  Case  I,  with  the 

result 

-'"    -  -* 


If  we  take  .r  —  0  as  the  line  at   inlinitv  and  use  nonhomogeneous 
('artesian  coordinates,  we  have,  tor  the  distance  between  two  points 

('..y)*ml  (•*•',/>,      j=      ,—^-ZT^M 


and    for    the    angle    between    the    two    lines     </./•-}-  /-//  +  c  =  0    and 
«'•<•  +  /''//  +  <•'  =  <>, 

cosli  0 

('onsider  now  anv  ti\ed  point  in  the  plane.  I;or  eonvonieneo  let 
it  be  the  origin  < >.  Through  (>  go  two  lines  of  the  pencils  defined  bv 
the  fundamental  conic:  that  is.  two  lines  drawn  to  the  fundamental 
points  at  inlinitv.  The  equations  of  the>e  lines  are  ./' ± // —  0 
(Fig.  4'J).  Thev  di\ide  the  plane  intot\\'o  regions,  which  \\c  mav 
mark  as  shaded  and  unshaded.  It  a  point  ( ./•,  in  hes  in  the  unshaded 
region,  jc"  -if"  ,-•  'I;  and  il  n  lies  in  the  shaded  region,  ./•"  -  i/~  <.  (I. 
(  i  'iisei  |  iieiit  lv,  distances  measured  ITOIII  <>  are  iman'inarv  in  the 


PRO.)  K<TI  VE   MEASi:  RKM  KNT 


11!) 


shaded  region  ;ui<l  real  in  the  unsluuled  region.  The  boundaries 
between  thr  two  regions  are  lines  ot  length  /.ero.  The  locus  of 
points  equidistant  from  a  are  equilateral  hyperbolas  j.'~—y~=k. 

A  line  '/./•  +  l>t/  —  0,  passing 
through  ( >,  is  in  the  unshaded 
region  it  <r  —  //' <  0  and  in  the 
shaded  region  if  <r—  />" ,  •  0.  Hence 
an  angle'  with  its  vertex  at  <>  is 
real  if  both  sides  are  in  the  shaded 
region  or  both  sides  in  the  un- 
shaded region,  and  is  imaginary 
if  one  side  is  in  the  shaded  region 
and  one  side  in  the  unshaded 
region.  A  line  through  ()  which 
is  not  a  line  of  /.ero  length  makes 
an  infinite  angle  with  each  of  the 
lines  of  /ero  length.  The  two  lines  of  /.ero  length  make  an  inde- 
terminate angle  with  each  other.  In  this  respect  as  in  other  wavs 
they  are  analogous  to  the  minimum  lines  in  a  Euclidean  plane. 

These  properties  are  of  course  the  same  at  all  points  of  the 
plane.  They  make  a  geometry  which  differs  widely  from  the 
geometry  of  actual  physical  experience.* 

*  This  L'cninetry  h;is  recently  trained  new  interest  because  of  its  oeeurreiiec 
in  the  theory  of  relativity,  ('f.  \Vil>oii  ami  Lewis,  "The  Spare-Time  Manifold 
of  Relativity,"  I'rui'ftdinyx  of  the  ^inurican  Araili-in/j  <>f  *lrtx  iital  >ViYnrc,s  (I'.U'J), 
Vol.  XLVII1,  .No.  11. 


CHAPTER  VIII 

CONTACT  TRANSFORMATIONS  IN  THE  PLANE 

51.  Point-point  transformations.    Consider  now  the  transformation 
defined  by  the  equations 


(1) 


where  s}.  .r.,,  .r,  and  .r[,  .r.',  ./•.'  are  point  coordinates  and  f^,  in  ,  fs  are 
homogeneous  functions  \vhirh  are  eoiitinuous  and  possess  deriva- 
tives and  for  which  the  .lacobian 


'.'•.         '.':         '.', 
(~'\        CJ',       '•>', 

cL       r/,       r/!_, 
r./-        r./',       r./\ 

'/':          '/'i          ^ii 
(  ./' 


does  imt  identically  vanish. 

liv  the  transformation  (1)  a  point  .r,  is  transformed  into  one 
or  more  points  ./•'.  with  possible  exceptional  points.  Owing  to  the 
hypothesis  as  to  the  .Jacobian,  (Mjuations  (1)  can  in  general  be 
solved  for  ./;,  and  any  point  ./•'  is  therefoi'e  the  transformed  point 
ot  one  or  nioie  points  ./• .  \vith  possible  exceptional  points. 

('on>ider  no\v  a  point  M  and  its  transformed  point  M'.  If  then- 
is  more  than  one  transformed  point,  we  will  !ix  our  attention  on 
one  only.  If  M  describes  a  curve  <•  delhied  bv  the  equations 

./.      <£,(/),      .'•.,=  (/).(/).      .r^  &,(>}.  C2) 

the   point    .!/'  describes  a  curve  /•',  the  e(|iiatioiis   ot    wliieli   may  be 

found    b\p    subsi  nut  ing   from   (  "J  )    into   (  1  ).     '!  he  direction    ot    <•  at 

M  i--   deiei'iiiiiied    b\    ./•,  ./' ,,  ./'..   and   '/./',.  '/./•,.  •/./-.,   as   sho\\-n    in    (  \  ), 

:>1.     'I'll''  direction   of  /•'  at    .!/'   is  determined  in  the  same  manner 


CONTACT   TRANSFORMATIONS   IN  THE  1'LANK      121 

by  -/•(,  ./•',  ./•',  and  <lj\,  </./•„',  <lj-[.  These  latter  six  quantities  are 
determined  by  tlie  former  six,  and  lieiire  the  direction  of  <•'  at  a 
point  .17'  is  determined  by  the  direction  of  <•  at  M.  From  this 
follows  the  theorem 

If  (tt'i>  <-f//wx  <•  ami  <\t  iirt'  ftiiii/i'tit  (it  a  point  M,  tin1  trdnxfonntnl 
cun'i'x  <•[  iiiLil  '•',  <//•«'  t<in<jt'nt  at  tltf  trunxfurintfd  point  M'. 

For  this  reason  the  transformation  (1)  is  called  a  Contact 
ti'dnxtvi'nmttoii. 

If  the  transformation  (1)  is  expressed  in  nonhomogeiieous 
Cartesian  coordinates,  it  becomes 

*'=/,(>,  //), 
!/'=/,(•'''  .'/)• 

Now  let  p  be  the  direction  —  —  of  a  curve  traversed  bv  the  point 

<lr         ,  , 

(./•,  //)  and  let  //  bt-  the  direction  -~-f  of  the  transformed  curve. 
\Ve  have,  evidently, 


ct'n          cf, 
' 


CJT 

P  =—  : 

' 


The  three  equations 


t 
1'  ~ 


are  called  an  enlarged  point  transformation.  Thev  brin^  into  clear 
evidence  that  two  curves  with  a  common  point  and  a  common 
direction  are  transformed  into  two  curves  which  have  also  a 
common  point  and  a  common  direction. 

52.  Quadric  inversion.  An  example  of  a  point-point  transforma- 
tion as  defined  bv  (1),  ^-"'1,  has  alreadv  been  met  in  the  case  of 
the  collhieations. 


l'2'2  TWO-DIMENSIONAL  GEOMETRY 

As  another  example  consider  the  transformation 

p.r(=  .rrrs, 

p/  =  ,y;i,  (1) 

p3£=sXjZr 

These  equations  can  be  solved  when  neither  .r,,  .r,,  nor  xa  are 
zero  into  the  equivalent  equations 


(2) 


The  transformation  establishes,  therefore,  a  one-to-one  relation 
between  the  points  j-{  and  the  points  r'  witli  the  possible  excep- 
tion of  points  on  the  triangle  of  reference  A  />(''.  To  examine  these 
points  let  A  be  as  usual  the  point  0:0:1,  />'  the  point  0:1:0,  and 
C  the  point  1:0:0,  so  that  the  equation  of  J/>'  is  a\=  0,  that  of 
AC  is  jrn  =  0,  and  that  of  />'('  is  r.t=  0.  Then  from  (1)  any  point 
on  the  line  J/>  is  transformed  into  />,  any  point  on  the  line  AC  is 
transformed  into  (\  and  any  point  on  the  line,  f-ic  is  transformed 
into  A.  The  coordinates  of  either  A,  />',  or  <",  if  substituted  in  (1  ), 
give  the  indeterminate  expression  0:0:  0,  but  it  we  enlarge  the 
definition  of  the  transformation  by  assuming  that  (-)  holds  for  all 
points,  including  those  on  Alt,  AC,  and  IlC,  it  follows  that  />'  is 
transformed  into  the  entire  line  A/>,  C  is  transformed  into  the 
entire  line  A<\  and  A  is  transformed  into  the  entire  line  BC. 
Consider  any  straight  line  with  the  equation 


It  is  transformed  into  the  curve 

«M+  «XX+^X=0, 

which  is  a  conic  through  the  points  A,  />',  and  C.  In  fact,  the  point 
in  which  the  line  meets  A/I  is  transformed  into  /),  the  point  in 
which  the  line  meets  AC  is  transformed  into  C',  and  the  point  in 
which  the  line  meets  IiC  is  transformed  into  A. 

It  the  straight  line  passes  through  one  of  the  points  .1,  /»',  or  (\ 
the  conic  into  which  it  is  transformed  splits  up  into  two  straight 
lines,  one  of  which  is  a  side  of  the  coordinate  triangle  and  the 
other  ot  which  passes  through  the  Vertex  opposite  that  side.  In 


CONTACT   TRANSFORMATIONS    IN    THK   TLANK      }'2:\ 

particular,  consider  a  line  j^+  X./;,=  0  througli  A.  The  first  two  of 
equations  (1)  give  jr\  +  \j~'n_  —  0  for  all  points  except  the  point  A  : 
that  is,  any  point  except  A  on  a  line  through  A  gives  a  definite 
point  on  the  same  line.  The  point  .1,  however,  goes  over  into  the 
entire  line  j~3=  0. 

In  a  similar  manner  a  conic  is  transformed  into  ;i  curve  of 
fourth  order,  which  passes  twice  through  each  of  the  points  A,  //,  C, 
since  the  conic  cuts  each  of  the,  lines  J/>,  IK',  ('A  in  two  points. 
If,  however,  the  conic  passes  through  one  of  the  points  A,  />',  <\ 
that  point  is  transformed  into  a  side  of  the  coordinate  triangle, 
and  the  curve  of  fourth  order  must  consist  of  that  side  and  a 
curve  of  third  order. 

In  particular,  a  conic  through  A  hut  not  through  />'  or  ('  is 
transformed  into  the  line  IK'  and  a  curve  of  third  order  through 
//and  ('.  A  nondegeiierate  conic  through  /.'and  Cand  not  through 
A  is  transformed  into  two  lines  Ah'  and  AC  and  a  conic  through  II 
and  (',  hut  not  through  A.  Finally,  a  nondegenerate  conic 
through  the  three  points  A,  />',  C  is  transformed  into  the  three  sides 
of  the  triangle  of  reference  and  a  straight  line  not  through  its  ver- 
tices. These  results  mav  all  lie  seen  directlv  or  verified  analytically. 

Bv  placing  ./•[=./•;  in  equations  (1)  the  locus  of  lixed  points  of 
the  transformation  is  found  to  be  the  conic 


which  passes  through  II  and  ('  and  is  tangent  to  AH  and  AC. 

It  is  not  difficult  to  show  that  each  point  /'of  the  plane  is  trans- 
formed into  a  point  /''  in  which  the  line  AT  cuts  the  polar  of  /' 
with  respect  to  tin;  fixed  conic. 

This  transformation  is  called  a  ^u/ii/ric  hn'erxiufi  to  distinguish 
it  from  the  circular  inversion,  or  simply  inversion,  discussed  in  the 
next  section. 

EXERCISES 

1.  Trove  the  statement  in  the  text  that  the  point  /'  is  transformed 
into  tin-  point  in  which  .!/'  cuts  the  polar  of  /'  with  respect  to  the 
fixed  conic.  Hcnee  sho\v  that  /'  and  /''are  harmonic  conjugates  to  the 
points  in  which  /'/''  cuts  the  conic. 


124  T\V<M)IM  KX  S  ION  A  L  G  EOM  ET  11  Y 

3.   Study  tin1  trans  format  ions 


i 


(2)    p.*-,'  =  .r  ,.,-,, 


p.r.   =  ./', 


=  .r.r, 


53.  Inversion.  Tlie  transformation  (1)  of  §  ">2  lias  particular 
interest  and  importance  \vhcn  the  points  /.'  and  ('  an>  tlic  circle 
points  at  infinity.  We  may  then  place  ./•.,=  /,  ./•  =  .r  +  ''/A  •''.,  =  •''  —  '// 
and,  usinijf  Cartesian  coih'dinates,  writ*'  the  transformation  in  the 


p  (-///)  =  (./•-  / 

pt'=jr+ifr 

or,  what  is  the  same  thing  in  nonliomogeneous  form, 


I>y  this  transformation  a  one-to-one  relation  is  established 
between  the  points  (./•,  // )  and  (./•',  //' ),  with  the  exceptions  that  the 
origin  corresponds  to  the  line  at  infinity,  and  conversely,  and  that 
each  of  the  circle  points  at  infinity  corresponds  to  the  minimum 
line  joining  it  to  the  origin,  and  conversely.  The  circle  ./•'-(-//"--  1 
is  fixed.  Any  point  of  the  fixed  circle  is  transformed  into  a  point 


CONTACT   TRANSFORMATIONS   IN   THE    PLANK      1  lM 

inside  that  circle,  and,  conversely,  in  such  a  way  that  if  <>  is  the 
origin,  /'  any  point,  and  /''  the  transformed  point,  (>/' .  (>/'' =  1. 
The  transformation  is  called  an  inri-rxion  with  respect  to  the  unit 
circle,  or  a  transformation  by  r>'fi/>ro<-ii/  ridlfii*  witli  respect  to 
that  circle.  The  origin  is  called  the  ft-iifer  of  inriTxi»n,  and  the 
fixed  circle  the  <•//•<•/<•  ,,f  ini',Txi'>n. 

Remembering  that  a  circle  is  a  conic  through  the  circle  points 
and  applying  the  results  of  the  previous  section,  we  have  the 
following  theorems: 

/.  A  ittrnii/Jtt  lint-  not  throni/h  (lie  renti-r  <>f  ////vrxA*//  /x  tnnutformetl 
info  n  ''i/'i'/t-  throni/h  tin-  i-i'ttti'r  <>f  inversion. 

II.  A   xtro-ii/ht   II in'  throiii/h   flit'  I'l'tittT  tit'  itii'i'rni"H   /x  transformed 
intu  itsrff  (iin<l  (In-  li/if  nt  infinit//'). 

III.  A  i'/ri'/i'  H"t  throut/h  fit*'  rmtfr  nf  hn'crainn  /x  tninsformetl  into 
it  <•/'/•'•/!'  tt'it   f/ir»i/<//i   (In-  i-i'iitiT  nf  ini'i'rxinH    (<n/>/  (Jn1   ft/'u   ninnniii/n 
Inii-x  through   tin-  i-fntt-r  <>f  invention"), 

IV.  A   a /'c/i'  tJifinif/Ji  (Jn-  t'l'iift-r  <>f  inrrrxion  /x  trftnttfornu'tl  into  a 
utrciif/hf  lii/i'  >/of  thmiujJi  f/if  ci'/ifi-r  nf  invention  (<m<1  (In-  f/r<>  i/iiiriinioti 
litn'x  llirowjli   (hi'   I'i'tifi'r  of  invention   <mi1  (In1   line   nt   infinity'). 

V.  A  font','  /x  trdnxformeil  in  i/?ni'r<il  info  n  <•><)•>••'  of  fourth  "/•'/•'/• 
fht'oi/i/h  tin'  <'ir/'/i'  fxiititx  nt  intiniti/. 

VI.  A   i'oiii<'  tln-otii/h   tin-    i-i'ntiT  of   'nu'i'fxion    /x    transformed   into 
,i    i-nri'i-    of   fltiril    onliT   throiiijJi    (In1    I'irrli-   points    (<nol    fh<:    Inn-    >it 
in  tin  it  I/ ). 

If  we  take  the  iionhoino^encous  form  ('2)  of  the  transformation 
and  apply  it  to  the  equations 

a i  +  1 1/  +  <• .-  0, 
n(.r  +/)+/)./•+  ••>/+/----  0 

we  readily  get  theorems  I-~IV  without  the  clauses  in  parentheses. 
It  is  in  this  simplified  form  that  the  theorems  arc  often  given,  but 
they  then  fail  to  tell  the  whole  story. 

Let  us  denote  bv  /  the  transformation  (1)  and  by  .17  the  trans- 
formation III,  ^1">.  Then  .)/  '  transforms  the  circle  ./•"+//"  k" 
into  the  unit  circle,  /  carries  out  an  inversion  with  respect  to  the 
unit  circle,  and  .17  carries  the  unit  circle  back  into  the  circle 
jr+i/--=k-.  The  product  of  these  three,  namely  .177.17  ',  which  is 


^o(;  T\\<>    DIMKNSIONAL   (JKOMKTKV 

the  transform  of  /  hv  -I/,  is  an  inversion  with  respect  to  the  circle 
x'jf-  //-  =  k~  and  is  represented  by  the  equations 

Irs 


•>  •  +//  =    •  ,  —  • 
./-  +  if 

It  is  cvitltMit  that  a  point  /'  is  transformed  into  a  point  /'',  where 
()/'.  ()!''  =  If2,  and  that  tlicorcins  I-VI  still  hold. 

It  \vc  dt-sirt-  an  inversion  with  respect  to  a  circle  with  center  (a,  /<) 
and  radius  /r.  we  may  transform  (8)  by  means  of  a  transformation 
which  carries  <>  into  (</,/<)•  The  result  is 

k-  (  .r  -  n  ) 
•'   —  "  =  .,  —  .—  r,  ' 


Ohviouslv   theorems  I    V  I   hold  for  (  •> ). 

If  the  inversion  (-)  is  written  as  an  enlarged  point-point  trans- 
•rmatioii  of  the  form  ('•'>).  §  ~>1,  we  have 


-  • 

j-  -  i/-  +  '2  f>.r>/ 

Fi'iim  tins  it  is  easy  to  compute  that  if  /-,  and  /'.,  are  the  slojtes  of 
two  curves  through  the  same  point,  and  if  />(  and  />'.,  are  the  slo]>es 
of  the  two  transformed  curves  through  the  transformed  point,  then 

/'.      /''     :   /',      /'a  . 
1  +  /''./'••       ^/'i/'a 

1  ln^  -hn\vs  that  the  alible  Itetweeii  two  curves  is  pi't'sei'ved  by 
t  !;e  t  raiisfonnatioii.  A  ti'ansf  on  nation  \\  hich  preserves  angles  is  said 
t'1  1"  ni t'irin-t/.  Ilenee  -///  i n r,  /'x/-,//  /x  n  i'nn1'<>nnnt  tr<t ny formation. 


CONTACT  TRANSFORMATIONS    IN    THE   PLANK      ll>7 

EXERCISES 

1.  Show  that  any  circle  through  a  point/'  and  its  inverse  point/'1 
is  orthogonal  to  the  circle  of  inversion. 

2.  Show  that,  a  pencil  of  straight  lines  is  transformed  by  inversion 
into  a   pencil   of  circles  consisting  of  circles   through  two  fixed   points. 
Study  the  configuration    formed    by   the   inversion   of  a  series   of  con- 
centric circles  and  the  straight  lines  through  their  common  center. 

3.  Show  that    parallel  lines  invert  into  circles  which  are  tangent  at 
the  center  of  inversion. 

4.  Show  that,  the  (Toss  ratio  of  four  points  collinear  with  the  center 
of  inversion  is  equal  to  that  of  the  transformed  points. 

5.  Show  that  a  point  /'  and  its  inverse  point  /''  are  harmonic  con- 
jugates with  respect  to  the  intersections  of  the  line  /'/''  and  the  circle 
of  inversion. 

6.  If  a  circle  is  inverted  into  a  straight  line,  show  that  two  points 
which  are  inverse  with   respect  to  the  circle  go  into  two  points  which 
are  svmmct rical  with  respect  to  the  line. 

7.  Study  the   real    properties   of  an    inversion   with    respect  to  the 
imaginary  circle  .>•"  -f-  //-  =  —  1. 

8.  Show  that  an   inversion  is  completely  determined  by  two  pairs 
of  inverse  points. 

9.  From  the  theorem  "four  circles  can  be  drawn   tangent  to  three 
given  lines"  prove  by  inversion  the  theorem  "four  circles  can  be  drawn 
tangent  to  three  given  circles  which  pass  through  a  fixed  point." 

10.  From  the  theorem  "two  circles  have  four  common  tangent  lines" 
prove  by  inversion  the  theorem  "through  a  given  point  four  circles  can 
be  drawn  tangent  to  two  given  circles." 

54.  Point-curve  transformations.  Consider  nmv  a  transformation 
defined  by  the  equation 

/••(.rr  ./•,,  .r:;,  ./-;,  X,  r.;)=0,  (1  ) 

\yhere  ./;  and  ./•'  are  point  coordinates  and  /•'  is  a  function  homo- 
geneous in  both  ./',  and  ./•',  con t  unions  in  bot  li  set  s  of  these  variables, 
and  possessing  derivatives  \vith  respect  to  both. 

Let    .!/  be  a  point   with  the  coordinates  ;/,-     If  these  coordinates 

are  substituted  for  ./',  in  (  1  )  and   held  fixed,  the   result  in '_;"  equation 

is  that    of  a   curve   which   we   call    an    ///'-curve,   the   equation    bein^ 

/•'(//,,  //,,,  //.,,  ./•;,  /•',  J\}-    <»,  CJ) 

and  we  say  that  t/if  i><>/n/   M  /*  //•<///*/'«/•///<'</  inf'.>  thr  ///'-<•///•/•<'. 


I'JS 


TWO   1>I  M  KNSK  >N  AL  ( J  KOM  ETK  V 


We  >hall  make  the  hypothesis  that  these  ///-curves  form  a  two- 
parameter  familv  of  curves  such  that  one  curve  of  this  family  goes 
through  anv  '^iven  point  in  any  given  direction. 

Let  A"  he  a  point  with  the  coordinates  z[.    This  point  will  lie  on  the 


and  all  values  of  the  ratios  //,://.,://.,  which  can  be  determined 
from  equation  <  :'. )  will,  if  used  in  ( "2),  determine  an  ///-curve 
through  A''.  These  values  ot  //(,  how-  ,]; 

ever,  are  given  bv  anv  point  .17  which 
lies  on  the  curve 

/••(./-,,  .r,,,  .r,,  ?[,  :      ,;•;>=  0.        (4)      ,^ 

('all  anv  curve  defined  bv  equation 
(4)  a  /'-curve.  We  have,  then,  the 
following  result  :  FIG.  43 

All  fmiiitx  .17  irliii-li  Hi'  on  a  k-eurve  are  transformed  into  m1 -curves 

/r/i/i'/t   pax*   t/tl'<H(<//t    it  jiniiit    /\  '   (  I' ig.  4-)). 

We  can  sav.  then,  that  (In'  k-furve  i*  transformed  into  a  point  I\' . 
In  fact,  the  tMiuatioii  of  a  /"-curve  is  found  by  holding  .r\  constant 
in  (1  ),  just  as  the  equation  of  an  ///'-curve  is  found  by  holding  .r 
constant  in  the  same  equation. 

It  is  further  evident  that  all  /c-ci/rrc.^  tr/u'rh  jxixx  throwr/li  a  point  M 
iin1  transf nnned  into  j><it/it*  l\ '  irlrieJi  lit'  on  the  cnrt'e  ///'. 

If  anv  proof  of  this  is  necessarv,  it  mav  be  supplied  by  noticing 
that  eijuation  (3)  is  the  condition  that  M  should  lie  on  /-  and 
that  l\'  should  lie  on  ///'.  ^ 

( '( msider  now  anv  eu  rve  <•, 
not  a  /'-eur\(',  delined  bv  the 
equal  i<  ins 


Fie..   H 


'I  he  ///'-curves  corresponding  to 
points     .17     on     i-     form     a     one- 
parameter  I'amiK'  of  curves  wliieli  in  general   have  an  envelope;  '•', 
and   the  <•///•/•,•  ,•  /'.-<  *///',/  /,,  /,>•  tranxfnrmeil  into  ///<•  rnr/'i'  /•'. 

To  follow   this  analytically   let    M,(J\,  :''.,,  -/',)   TKig.  41)  l>e  the 
point    on    '•  corresponding    to    the    value   A.    of    X,   and    let    .17,   be 


CONTACT   TRANSFORMATIONS    IN    TIIK    PLANK 

the  point  corresponding  to  the  value  A.  -|-  AX,  the  coordinat 
.)/,  being  .r}+  A./-J,  ./•„-(-  A./-  .,,  ./-^  f  A./-.t.  The  t\vo  points  .)/  and  .)/ 
are  transformed  into  >/>[  and  ///',,  which  intersect  in  a  point  A",  the 
coordinates  of  which  are  given  l>v  the  e(|uations 


where    the  values  of   ./•_  and   A.r,   are    (n    he    taken    from   (•">).     The 
point    A"'  corresponds  to  a  /--curve  through  J7   and   M  . 

Now  let  .!/,  approach  .'/,.  The  curve  m'.,  approaches  the  curve 
;//(,  and  the  point  l\'  approaches  a  limiting  point  '/''  the  coih'dinatcs 
of  which  are  given  hy 


r  /•'  r  /•'  ,<-/•'  (7  ) 

-  7./-  +         '/./'  -I-          <Lr  —  0, 
CJ-j  f  ./'.,  r.r 

wliere  the  values  of  ./-,.  and  </./•.  are  to  lie  taken  from  (•">). 

The  point  T'  is  obviously  the  transformed  point  of  /,  a  /'-curve 
taaigent  to  '•  at  J^.  'The  locus  c»f  7''  is  the  curve  '•',  which  cor- 
responds to  c. 

Equations  (7)  furnish  a  proof  that  <•'  is  tangent  to  ///'  at  T'. 
For,  by  differentiating  the  iirst  of  these  equations  and  taking 
account  of  the  second,  \\  c  have 


which,  as  in  ^  :>1,  determines  the  direction  of  <•'.  l.ut  this  is  jus; 
the  equation  \\hich  determines  the  direction  <>t  ///[.  I  he  direct  n  in 
ol  '•'  is  thus  determined  at  the  point  7''  l>v  the  direction  ol  ///',.  \\ 
is  therefore  determined  l>v  the  point  M  and  the  curve  /.  the  latter 
being  determined  bv  the  direction  of  <•.  lleiiee  tn'n  ruri'r*  <•  //•///< 

<//•>'   f'Dt'/rtlf    ili'i'   t  r<l  nxt'i'l'/m  <l    ilit'i   til',,    fl/t'l'fX   <•'   It'/it'i-Jl   '//•••    t'Uli/i'llf.       I   lie 

transformation  is  therefore  called  a  rmitiii't  tr<inx1't>rnnttt"n. 

Suppose   now  that    the  transformation   (  1  )   is  expressed   in   non- 
homogeneous  ('artesian   coi'ii'dinates  b\    the  equation 


1;|()  TWO   DIMKNSIONAL  GKO.MKTKV 

and  let  />  be  the  slope  --  of  anv  curve  <;  and  //  the  slope  --—.  of 
r/.r  df' 

the  transformed  curve  <•'.    Then  equations  (<>)  and  (S)  are  replaced 
in   the  present  coi'irdinates  by 

cF         cF      A 

-  +  />  4     =  0, 


. 

termine  />  and  //  when  .r,  //,  ./•',  and  //'  are 
ons       ritten  toether 


which  enable  us  to  determine  />  and  //  when  .r,  // 
known.    The  last  three  equations,  written  together, 

F(j;  //,  .r',  /  )  =  0, 


r.r  r/y 

^+X^  =  », 

r.r'  ry 

are    called    an    enlarged    point-curve    contact    transformation.     If 
solved    for  r',  //',   and  //   they   may   be   written   in   the  form 

*'=/!<>,    /A/')' 

y'=f*(r>y>p)>  00) 

//  =  ./';!(.r,  //,  ;>). 
If,  then,  the  point  (  .r,  //  )  desoribps  the  curve  r  —  _/'  (Xs),  //  =f.,(  X), 

we  have  />  -  '  J     "    ,  and  equations  (1<I)  Lfive  the  transformed  curve 

/,(.M 

expressed  in  terms  ot  the  pai'ameter  X. 

An  example  ot  a  point-curve  transformation  is  found  in  the  cor- 
relation^ already  discussed,  since  the  eiiuations  (1  ),  ^-1-,  mav  be 
written  in  the  form 


llrre  the  ///-curves  and  the  /--curves  are  straight  lines.  If  .r; 
de^crilirs  a  ciir\'e  '-.  the  Mrai'_dit  line  ///'  envelops  the  transformed 
ciirvr  '•'.  If  tin-  correlation  is  expressed  in  Cartesian  coordinates, 
it  is  ivadilv  jmt  into  the  form  (1"). 


CONTACT   TRANSFORMATIONS    IN   THE   PLANE      131 

EXERCISES 

1.  Express  the  gem-nil  correlation  in  the  form  of  equations  (10). 

2.  Place  in  the  form  of  equations  (10)  the  polarity  l>v  which  a  point 
is  transformed  into  its  polar  line  \vith  respect  to  the  circle  j-~  -\-  if  =  1. 

3.  Kind  the  curve  into  which  the  parabola  if—  <ts  is  transformed  bv 
the  polarity  of  Ex.  -. 

4.  Show  that  the  curve  into  which  the  circle  (./•  —  /')"-}-('/  —  /.')"=  r~ 
is  transformed  bv  the  polaritv  of  Ex.  !J  is  a  conic,  and  state  the  con- 
ditions under  which  it   is  an  ellipse,  a  parabola,  or  a  hyperbola.     Find 
the  focus  and  directrix  of  the  conic. 

5.  1'rove  that  by  any  polarity  the  order  and  the  class  of  the  trans- 
formed curve   is  equal  to  the  class  and  the  order,  respectively,  of  the, 
original  curve. 


6.  Study  the  transformation 


i      .'/ 

'-;-*• 


and   find   the   curve   into   which    the    circle   j--+  >f  =  3    is    transformed 
by   it. 

7.  Express  in  the  form  of  equations  (10)  each  of  the  types  of 
correlations  f^iven  in  $ -I-  and  study  them  from  this  standpoint. 

55.  The  pedal  transformation.  As  another  example  of  a  point- 
curve  transformation  we  shall  use  homogeneous  Cartesian  coordi- 
nates and  take  the  equation 

(.r'--f//'-)/-.rVV  -if't't/=().  (  1  ) 

II  we  take  M  as  any  point  (./•;_//:  t),  the  corresponding  ///'-curve 
is  in  general  a  circle  constructed  on  the  line  <>M  as  a  diameter. 
Ivxceptional  points  are  the  origin  and  the  points  at  intinity.  Il  '/ 
is  the  origin,  the  circle  becomes  the  two  minimum  lines  through 
the  origin.  If  .17  is  a  point  at  infinity,  not  a  circle  point,  the  circle 
t/i'  splits  up  into  the  line  at  intinity  and  a  straight  line  through  O 
perpendicular  to  o.M.  If  .)/  is  a  circle  point  /,  the  circle  in'  splits 
up  into  the  line  at  infinity  and  the  minimum  line  <>/. 


13:>  TWO-DLMEXSIOXAL  GEOMETRY 

The  /'-curve  corresponding  to  a  point   A"'  is  in  general  a  straight 

i  ~  i  o  o 

line  through  A"' and  perpendicular  to  <>/\'.  Exceptions  occur  when 
A"'  is  the  origin  or  one  of  tlie  circle  points  at  infinity,  in  which 
cases  the  /--curve  is  indeterminate.  It'  A"'  is  any  point  on  the  line 
at  intiiiitv  hut  not  a  circle  point,  the  /--curve  is  the  line  at  inlinitv. 
It  A"  is  mi  a  minimum  line  through  o,  hut  not  at  infinity,  the 
/•-curve  is  the  other  minimum  line  through  ().  A  /Mine  does  not 
in  general  pass  through  <>  or  the  circle  points  at  inlinitv. 

( 'on verselv,  anv  straight  line  which  does  not  pass  through  the 
origin,  and  is  neither  the  line  at  inlinitv  nor  a  minimum  line,  is  a 
/•-line,  the  point  A'  heing  the  point  in  which  the  normal  from  () 
meets  the  line.  This  may  he  seen  hy  comparing  the  equation 
".''  +  I'll -t-i't  —  ()  with  (  1  ),  thus  determining./'://':  /'—  —tic:  —  lc: <r-f//J, 
which  is  the  foot  of  the  normal  from  (>  to  the  line. 

Take  anv  curve  <•.  The  tangent  /--curve  at  anv  point  M  is 
the  tangent  line  t,  and  the  point  '/''  is  the  foot  of  the  perpen- 
dicular li'oin  <>  on  T.  Therefore  the  tntmtfnnnetl  CHITC  <•'  <>f'  ami 
<•///•/•<'  !•  /s  flu1  /IK-UK  lit  tin1  Ji'ft  >>t  tin'  pei'jH'Hilit'ufurs  (Irairn  from 
tin'  »/•('////  fa  tin'  fiitii/ftif  liio'x  <>f  i'.  The  transformation  is  called 
the  i»-<l<il  (I'linxforiniitioti,  and  the  point  (>  is  the  urii/iu  of  the 
transfi  irmat  ion. 

If  the  pedal  transformation  is  expressed  in  Cartesian  coordi- 
nates as  an  enlarged  point-curve  transformation  of  the  form  ('.'), 

vj  .)  \.  it  becomes 

,'•'-+  //'---  ./•'./•  —  //'//  =  0, 


and   these  conations  can   he  solved   tor  ./•',  //',  and  y/,  giving 

(  //  -  ;*./•)/* 


CONTACT  TRANSFORMATIONS   IN  T1IK  PLANK      1  ;•}.') 

EXERCISES 

1.  If  (,>  is  the  pedal  t ransformat  ion   with   the  origin    <>,   /'  a  polarit\ 
with    respect    to    auv    circle    with    the    center    <>,    and    /,'    an    inversion 
with   respect    to  the   same   circle,   prove  the   relations    <J  -----  HI',   /'  -~  I!<1, 
It  =  (ll>. 

2.  Show  that  bv  a  peilal  traiist'orinatioii  a  parabola  with  its  focus  at 
the  origin  of  the  transformation   is  transformed  into  the  tangent  line 
at  the  vertex  of  the  parabola. 

3.  Show  that  l»v  a  pedal  transformation  an  ellipse  with  its  focus  at 
the  origin  of  the  transformation   is  transformed   into  a  circle  with  its 
diameter  coinciding  wit  h  the  major  diameter  of  the  ellipse.    State  anil 
prove  the  corresponding  theorem  for  the  hyperbola. 

4.  Find  the  curve  into  which  the  ellipse  -.,  -)-'.,  =  !   is  transformed 
by  a  pedal  transformation  with  its  origin  at  the  renter  of  the  ellipse. 

56.  The  line  element.  With  the  use  of  Cartesian  coordinates  the 
contact  transformations  may  be  looked  at  from  a  ne\v  viewpoint 
bv  the  aid  of  the  concept  of  the  /i/n-  </>  //n  /if.  A  line  element  mav 
be  defined  as  a  point  with  an  associated  direction.  More  precisely 
let  there  be  given  three  numbers  (./•,  //,  y>),  where  the  numbers 
./•and  if  are  to  be  interpreted  as  the  usual  ('artesian  coordinates 
of  a  point  in  the  plane  and  />  is  to  be  interpreted  as  the  slope 
or  direction  of  a  line  through  the  point.  Then  the  three  <{iianti- 
ties  taken  together  define  a  line  element.  A  line  element  mav 
be  roughly  represented  by  plotting  a  point  M  and  drawing  a  short 
line  through  M  in  the  direction  />,  but  this  line  must  be  con- 
sidered as  having  no  length  just  as  the  dot  which  represents  M 
must  be  considered  as  without  magnitude.  There  are  -/_'''  line 
elements  in  the  plane  out  of  which  \\  e  may  form  a  one-dimensional 
extent  of  line  elements  by  taking  ./',  //.  and  //  as  functions  of  a 
sinle  arameter:  thus. 


There  are  two  types  of  oiie-dinieiisioiial  extents: 

Tvi'K  I.  The  fund  ions  f^  (  \  )  and  _/',(  X")  mav  reduce  to  constant 
In  this  case  the  one-dimensional  extent  consists  of  a  fixed  poii 
with  all  possible  directions  associated  with  it. 


}:}.{  TWO    D1MKNSIONAL  CEOMKTKV 

Tvi'K  II.  The  point  (  /',  // )  may  describe  a  curve  the  equations 
of  which  are  the  lirst  two  of  (1  ).  Then  the  third  equation  of  (1) 
associates  \\ith  every  point  of  that  curve  a  certain  direction. 

It  is  ob\  ioiislv  convenient  that  the  direction  associated  with  each 
point  of  the  curve  should  be  that  of  the  tangent  to  the  curve.  The 
necessary  and  sulh'cient  condition  lor  this  is  that  by  virtue  of  (1  ) 
We  should  have  </./'  /»///  =  It. 

A  one-dimensional  extent  of  line  elements  defined  by  equation  (1  ) 

shall  be  called  a  mil"//  of  line  elements  when  it  satislies  the  con- 
dition </./•  -jnlif  =  ().  It  is  evident  that  the  lirst  tvpe  of  extents 
always  satisties  this  condition  and  that  the  second  type  satisties  the 
condition  when  the  direction  of  each  element  is  that  of  the  curve 
on  which  the  point  of  the  element  lies. 

Two  unions  of  line  elements  have  <-<>nt<i<-t  with  each  other  if  they 
have  a  line  element  in  common.  Two  unions  of  the  first  type  have 
contact,  therefore,  when  they  coincide ;  one  of  the  lirst  type  has  con- 
tact witli  one  of  the  second  when  the  point  of  the  lirst  lies  on  the 
curve  of  the  second:  and  two  elements  of  the  second  type  have 
contact  when  their  curves  are  tangent  in  the  ordinary  sense. 

Any  transformation  of  line  elements  detined  by  the  equations 


where  the  functions  are  bound  by  the  condition 


where  p  is  not  identically  x.ero,  is  called  a  <-u/if,i,-t  tr<inx1\irmcit'mn. 

It  is  dear  that  by  such  a  transformation  a  union  of  line  ele- 
ments i>  transformed  into  a  union  of  line  elements  and  that  two 
unions  which  arc  in  contact  are  transformed  into  two  which  are 
in  contact. 

'I  he  enlarged  point-point  transformation  (  '•*>  ).  >  ;~>1,  and  the 
enlarged  point-curve  transformation  (It),  x  -~  4.  arc  cases  of  the 
general  contact  transformation  (  -  ).  In  tact,  any  contact  trans- 
formation may  be  reduce*]  to  one  of  these  cases.  'I'o  show  this 
let  u<  proceed  to  deduce  from  (  •_'  >  ((piations  \vhidi  are  Iree  from 
/'  and  /''.  I  \\  o  cases  on]  v  can  occii  r. 


CONTACT   TRANSFORMATIONS   IX   THE    1'LANK      1  :J.j 

('ASK  I.    Tlu-  first  two  equations  in  ('2)  niav  cadi  he  fret-  t'nnn  /-. 
Then  (-([nation  (  -I  )  skives  the  condition 

rf,  rf.,  if-f  ff 

-•-  <Lr  +  -^  <lif  -  >>'  -    '  */./•  -  //  -    '  ,lif  =  p  (  //;/  -  ;»//./•  ), 

o1  r//  rr  ct/ 

which  must  he  true  for  all  values  of  the  ratios  </./•  :  </_//.     I  leiice  \ve  have 

cf.,         ,'cf. 

-  r  •    =  />< 
1  y       (  // 


C.C  C.f 

whence,  hv  eliminating  p  and   solving   for  y',  \ve   have   the   result 
that  the  contact   transformation  (2)   is  in   this  case  of  the  form 


^  './•  r  y 

which  is  exactly  that   of  (  o  ),  £  ">1. 

I>v  this  t  raiisfoi'inat  ion  any  oiic-ilimensional  extent  of  line  ele- 
ments which  form  a  union  of  the  first  tvj>e  is  transformed  into  a 
union  of  the  first  tvpe,  and  anv  union  of  the  second  tvpe  is  tran>- 
formcd  into  a  union  of  the  second  tvpe. 

(  'ASF.  II.  At  least  one  of  the  lirst  \\\n  cijuat  ions  in  (  '1  )  contains  //. 
It  is  then  jiossihle  to  tind  one,  hut  only  one,  (.-(juation  fn-e  from 
//  and  //.  Let  that  equation  lie 

/'(./•.  //,  ./•',  //'  )  —  0. 
From   this  equation   we   tind 

(  /•',         f  /•'  /  /•'      ,      f  /•'       . 

'l.i-  +       </>/  +      ,«/./•+      ,  <1<i  =0, 

(  ./'  '    'I  '  •>'  '   <l 

which  mu>t   he  identical  \\ilh  (•'•).     \\\  comparison  we  find 


p/> 


136  TWO-DIMENSIONAL  GEOMETRY 

from  which  y  and  />'  can  be  found,  with  the  result  that  the  contact 
transformation  (^-)  ran  in  this  ease  he  put  into  the  form 


ex  cy 

cF        ,cF 

—,+P.    ,  =  °> 
ex'          et/ 

whieh  is  exactly  that  of  (9),  §  54. 

r>v  this  transformation  any  union  of  the  first  type  is  transformed 
into  a  union  of  the  second  type,  each  element  of  the  former  being 
transformed  into  an  element  of  the  latter. 

As  an  example  consider  the  transformation 


- 
VI 


If  written  in  the  form  (  f>  )  this  becomes 


The  geometrical  meaning-  of  these  equations  is  simple.  Any  line 
element  (  r,  //,  //)  is  transformed  into  a  line  element  (./•',  ?/',  //  )  so 
placed  that  the  point  (./•',  //')  is  at  a  distance  k  from  the  point  (./•',  //'). 
and  the  line  joining  (a*',  //')  to  (^-,  //)  is  perpendicular  to  the  line 
element.  A  transformed  line  element  is  parallel  to  the  original 
element.  Otherwise  stated,  each  line  element  is  moved  parallel 
tn  itself  through  a  distance  /••  in  a  direction  perpendicular  to  the 
direction  of  the  element.  Kach  line  element  is  therefore  trans- 
formed into  two  line  elements.  A  union  of  the  first  type,  consist- 
ing ot  line  elements  through  the  same  point,  is  transformed  into  a 
union  consisting  of  the  line  elements  of  a  circle  with  that  point  as 
a  center  and  ;i  radius  /-.  Any  curve  <•  is  transformed  into  two 
curves  parallel  to  r  at  a  normal  distance  /-  from  <•. 

'I  his  transformation  is  sometimes  called  a  ifi/iiti>in,  suggesting 
that  each  point  ot  the  plane  is  dilated  into  a  circle. 


CONTACT  TRANSFORMATIONS   IN    THE  PLANK      137 

EXERCISES 

1.  Show  that  the  transformation 

*'=;>, 

//'=  •''/'  -  //i 
//=  .'•. 

is  a  contact  transformation  and  stiulv  its  properties. 

2.  Show  that  the  transformation 

SC'=:jr  +  »JJ, 

.'/'  =  //+/'"» 
J''  =  1', 

is  a  contact  transformation  and  study  its  properties. 

3.  Show  that  any  differential  equation  of  the  form/Mr,  //,  -'   )  =  0 

may  be  written  in  the  form  j'(s,  //,  />)  =  0  and  considered  as  defining  a 
doubly  intinite  extent  of  line  elements.  To  solve  the  equation  is  to 
arrange  the  elements  into  unions  of  line  elements.  In  general,  the  solu- 
tion consists  of  a  familv  of  curves.  Anv  union  formed  by  taking  one 
element  from  each  curve  of  a  familv  is  a  singular  solution.  Note  that 
an  equation  j\.i\  //)  —  ()  can  also  be  interpreted  in  this  way,  and  that 
the  family  of  solutions  consists  of  points  on  the  curve ,/'(.'',  </)=  0  with 
all  the  line  elements  through  each,  while  the  singular  solution  is  the 
curvey'i ./•,  // )  =  0  with  its  tangent  elements. 

4.  Study  the  differential  equation  // — //./•  =  0  in  the  light  of  Kx.  .'!. 
Show  that   the  singular  solution  is  the  one-dimensional  extent   of  line 
elements  which  consists  of  all  elements  through  the  oriirin. 


5.    Aiiplv  to  Kx.  4  the  dilation  ./•'=./•  —  .   //  =  //  + 

1        »  v  ,  V*  »' 


]'  =  !•'.   Show  that  the  different  ial  equation  becomes  //' — //./•'—  v  1  -f-/''"=  0. 
"\\hat  becomes  of  the  singular  solution  and  the  familv  of  solutions'.' 

G.  Study  Clairaut's  equation,  _//  =  //./•  -f-_ ;'( // 1.  by  the  method  of 
K\.  .'>  and  show  geometrically  that  the  familv  of  solutions  consists  of 
the  straight  lines  >/  =  <•./•  +/{<•).  What  is  the  singular  solution  '.'  Apply 
to  the  variables  in  the  equation  the  transformation  ./•.<•'  -f-  ////'=  1  and 
determine  the  effect  on  the  equation  and  its  solutions. 


CHAPTER  IX 

TETRACYCLICAL  COORDINATES 

57.  Special  tetracyclical  coordinates.  We  shall  discuss  in  this 
chapter  a  >ystem  of  coordinates  especially  useful  for  the  treat- 
ment of  the  circle.  These  coordinates  are  not  dependent  upon  the 
('artesian  coordinates,  though  they  are  often  so  presented.  ( )n  the 
contrary  they  mav  be  set  up  independently  by  elementary  geometry 
for  real  points  and  then  extended  to  imaginary  and  infinite  points 
in  the  usual  manner.  It  is  therefore  not  to  be  expected  that  the 
geometry  in  the  imaginary  domain  and  at  y 
infinity  should  a^ree  in  all  respects  with 
that  obtained  by  the  use  of  Cartesian 
coordinates. 

The  coordinates  we  are  to  discuss   are       -N 
called    tctraevelical    coordinates,    and    we 
begin,  for  convenience,  with  a  special  type. 

Let  <>X  and  <>Y  (Fig.  4.*))  be  two 
straight  lines  of  reference  intersecting  at 
right  angles  at  O,  and  let  /'  be  any  real  point  of  the  plane.  Let 
-Wand  SI'  be  the  distances  of  /'  from  o.\'  and  O  )'.  respectively, 
taken  with  the  usual  convention  as  to  signs,  and  let  <>!'  be  the 
distance  of  /'  from  <>,  taken  always  positive.  Then  the  special 
tetracvelieal  coordinates  of  /'  are  the  ratios 


=  <>r~:NI':  Ml':  1, 


(1) 


from    which    it    follows    that    the    (juantities    ./•    are    connected    by 
the   fundamental    relat  H  m 


It  is  ohyioii>  that  to  any  real  point  corresponds  one  set  of  coor- 
dinates and,  conversely,  to  any  real  set  of  the  ratios  ./^ :  ./•  :  ./-._:  ./^ 
\\hii-li  -ati-lV  the  relation  ('2).  and  for  whieli  ./•.  —  <),  corresponds 
one  real  point  /'.  \\V  extend  the  coordinate  system  in  the  usual 


TKTUACYCMCAL  COORDINATES  I:;1.) 

manner  hv  the  convention  that  any  set  of  ratios  satisfying  ( - ) 
shall  define  a  real  or  an  imaginary  point  ot  the  plane,  the  ratios 
0:0:0:0  being  of  course  unallowed. 

As  the  real  point  /'  recedes  from  o,  the  ratios  approach  a  limit- 
ing set  of  values  1:0:0:0.  To  see  this  we  write  equation  (1  )  in 
the  form 


or"   or~  oi'" 

cos  0    sin  6       1 


=  1  : 


where  6  =  the  angle  MO/'.  The  limit  of  the  ratios  of  .r  is  there- 
fore 1  :  0:0:0.  Hence  we  say  that  lij  (Jn-  HX>'  <>f  tin'  #jn>ri<t/  ti'trn- 
i'i/i'lii'nl  coordinates  t/it'  plain'  /x  regarded  <<*  having  <t  van/If  /•<<>/  /'"hit 
at  i»jiint>/.  This  point,  however,  is  not  the  only  one  which  must 
be  considered  at  infinity,  as  will  appear  later. 

58.  Distance  between  two  points.  Let  c  (/^ ://.,://.,:  //4 )  and 
/'  ( r^.  ./•.,:  .r^:  ./'4)  (Fig.  40)  be  two  real  points,  and  let  </  —  <T,  the 
distance  between  them.  Then,  by  trigonometry, 


where    0  —  the    angle    X<>/'    and    $.,=  the    angl 
the    definition    of   the    coordinates    and  y 

from    the    relations 

ol'  cos  0^-.=  ''-,      Or  sin  0}  =  ''  \ 
•' -i  -'4 

or  cos  0  =  ''--•>      ()('  sin  ft      //:s 
//*  //.. 

the  above  equation   can   be  written 


(  1  ) 


This  e(|iiation,  obtained   hv   the   use  of  real   points,  is  now   taken 
us  the  definition  of  the  distance  between  imaginary  points. 
Equation  (1)  can  be  written 

(?  —  (  -  ) 


140  TWO-DIMENSIONAL  GEOMETRY 

where  in  accordance  with  the  usual  notation  o>(.r,  i/')  denotes  the 
polar  *  of  the  form  ol^.r). 

From  (T)  it  appears  that  <7=x  when  //4  =  0  or  when  .r4  =  0.  ITence 
tlif  I"i-iix  <>f  the  point*  dt  infinity  ix  defined  by  the  equation  .r  =  0. 

Since  always  «  (J')=  ()'  tn(1  points  at  infinity  satisfy  also  the  con- 
dition ./•;+.<'/=  0,  from  which  it  appears  that  the  point  1:0:0;  0 
is  the  only  real  point  at  infinity,  as  we  have  already  seen.  The 
nature  of  the  locus  at  infinity  will  appear  later. 

59.  The  circle.  If  \ve  take  the  usual  definition  of  a  circle,  the 
equation  of  a  circle  with  center  //.  and  radius  /•  can  be  written  from 
(1  ),  £  ~>8,  as 

This  is  of  the  type 

and  the  relations  between  the  coefficients  <i{  and  the  center  and 
radius  of  the  circle  are  readily  found.  For  we  have  by  direct 
comparison  of  (1)  and  (2) 


From   these  and  the  fundamental  relation  //.,"  +  ,^3"—  //ltV4  =  0  we 
easilv  compute  the  following  values: 


=  —-<l\nv 
4  =  4  a,2, 

af+  a—  4  a  ^r 


anil  the  bilinear  furin  //fit-J'i?/A- 

i 

is  callnl  tlif  ]mlar  fni'in  "f  (1).    If  by  a  linear  transformation  of  thr  variables 
the  furin  (1)  is  transformed  intn 


its  tiolar  i.-  t  ran.-fnrmed  int. 


TETKACYCLK'AL  COORDINATES  141 

which  give  the  coordinates  of  the  center  ;m<l  the  radius  of  the 
circle  in  terms  of  the  coefficients  </.  of  equation  ('_'). 

These  results,  obtained   prinuirily  for  mil  circles,  are  now  <»vn- 

v  !~t 

erali/.ed  /'//  definition  as  follows: 

Ert'ri/  liin'itr  conation  of  1  hi' form  (2)  reprext>ntx  n  <-//v/»',  the  renter 
{tn<l  tlt>'  r<itl'nis  of  which  <ir>:  ijiven  /<//  equations  (-5). 

We  may  classify  circles  hy  means  of  the  expression  for  the  radius. 
For  that  purpose  let  us  denote  the  numerator  of  /••  in  ('•})  by  ?;  ('') : 

that  is'  >;(«)  =  ";+";- 4, vv  (4) 

We  make,  then,  the  following  cases: 

CASK  I.    77 (a)  =^  0.    X<>nKp>'<-iiil  <-f/-i-Ji'fi. 

,V///'<v.sv  .7.  f^  ^  0.  Proper  <'ir<-fi'N.  Kfpiation  (-)  is  reducihle  to 
(  1  )  and  represents  the  locus  of  a  point  at  a  constant  distance  from 
a  fixed  point.  Neither  center  nor  radius  is  necessarily  real,  hut  the 
center  is  not  at  infinity  and  the  radius  is  finite.  The  circle  does  not 
contain  the  real  point  at  infinity,  since  1:0:0;  0  will  not  satisfy 
equation  (2). 

Xnfiniitf'.  2.  <t  —  0.  Ordinary  sfnn'i/ht  l/m'.*.  The  radius  becomes 
infinite  and  the  center  is  the  real  point  at  infinity.  The  equation 
may  he  written,  by  ^  ^T,  in  the  form 

aaXi>  4.  f,^ff>  +  <r^  =  o,          (,/.;  +  ,/:f  -^  0) 

which,  as  in  ('artesian  geometry,  is  a  straight  line.  This  line 
passes  through  the  real  point  at  infinity.  In  fact,  the  necessary 
and  sufficient  condition  that  equation  (-)  should  be  satisfied  by 
the  coordinates  of  the  real  point  at  infinity  is  that  a  =  0.  llenee 
mi  nfli  i>ii  nf  fttriti'/ht  ///"'  nun/  /n'  tie  fin  I'll  >i*  >'  nttnupei'ial  fircL-  //'///••// 
im*vi'x  throitffh  the  real  }i»fnf  nt  iiifitiitj/. 

CASK  II.    7;0/)=0.    Sprriitl  cirri?*. 

Since  it~  -f-  <i-l  —  4  "!''.,.  the  coordinates  of  the  center  may  be  written 

//, :  .?/._, :  .'/, :  ,V4  =  -  2  ", :  ",, :  « . :  ~  -  -'r  (  ">") 

,v////,v/.sv  7.   a  ^  0.    /'"//// r //•-•/,  x.    The  radius  is  xero  and  the  coi'irdi- 

nates  of  the  center  are  those  of  a  point  not  at   infinity.    'I  lie  center 
may  lie  any  finite  jioint.     It   is  obyioiis  that    if  the  center  is  real,  it   is 
the  only  real  point  on  the  circle,  and  hence  the  name      point  circle. 
The  point  circles  do  not   pass  through  the   real   point   at    infinity. 


142  TWO-1>IMKNSIONAL  CKOMKTKY 

Rv  (-)<  $  .~>S,  the  equation  of  a  point  Circle  may  be  written 

(D(  .r.  y)  =  0, 

\vliere  <•«>(//)  0.  Comparing  with  (4),  we  see  how  the  equation 
7;  ( «/ ) -=  0  niav  be  deduced  from  «(//)=  0. 

.SW-.W.W  .'.  ^=0.  Sfwinl  xfrtiii/ht  lint'*.  The  radius  becomes  inde- 
terminate, and  the  center,  given  by  (4),  becomes  —  2<r:tf  :rt3=0, 
which  is  a  point  at  infinity.  The  special  straight  lines  pass  through 
the  real  point  at  iniiiiitv.  In  fact,  a  xpt'<-ial  atrdi'/ht  line  mm/  /<»' 
di  fin-  'I  <ia  <i  xjii-i-iiil  '•//•••/>'  >rhifh  pii.-iK,'s  through  the  real  point  at  infinity. 

We  have  seen  that  the  locus  of  all  points  at  infinity  is  .>•=<(. 
which  is  the  equation  of  a  circle  belonging  to  the  case  now  being 
considered,  and  with  its  center  at  1  :":":<>.  Hence  we  sav: 

Tlif  I'x-ux  at  in  fin  it  i/  i*  <(  npi'i-iiil  xt  rai'/ld  line  whose  center  ?x  the 
r<  <tl  pniiit  at  infinity. 

EXERCISES 

1.  ( 'oiisider   the    point   circle   •?.=  ".     Slio\v   that   it    is    made   up  of 
two  one-dimensional  extents   (''threads''^  expressed  l>v  the  equations 

.r  :  .r  :  .r  :./•.  =  0  :  1  :  -i-  /  :  A.  wliei'e   A   is  an  arbitrarv  iiai'auieter.    Sho\v 
i      j      x      -i  i 

that  these  threads  have  the  one  point  0:0:0:1  in  common,  but  that 
neither  can  be  expressed  bv  a  single  equation  in  tetraeyclical  coiirdi- 
nates.  Hence  note  the  difference  between  tliis  locus  and  that  expressed 
by  ./•-  -f-  //-  =  0  in  Cartesian  coordinates. 

2.  As  in  Kx.  1.  sho\v  tliat  the  s]ieeial  circle  .?-4  —  0  is  composed  of  two 
threads  having  the  real  point  at  infinity  in  common. 

3.  Examine  the  special  circles  r, -f-  /r.(  =  0  and  r,  —  i.r   =  0  and  show 
that  these  two  and  the  two   in    Kxs.  1  and   '2  are  made   up  of  different 
combinations  of  the  same  four  threads. 

4.  Show  that  any  special  circle  is  made  up  as  is  the  circle  in  Kx.  1. 

60.  Relation  between  tetraeyclical  and  Cartesian  coordinates.  I  f  we 
introduce  ('artesian  coordinates,  bv  which,  in  Fi'_r.  4-~>, 

.r:  y:  t=*OM:  Ml':  1. 

there  exists  for  anv  real  point  of  the  plane  the  following  relation 
between  the  special  tetraeyclical  coordinates  and  the  ('artesian 
coordinates:  ps^jr+fr 

p-'~,  =  J't, 

P-I-,  =  .'/', 

P-'\   --  <*• 


ATKS  l  }:; 

These  C([uations,  derived  for  real  points  of  the  plane  at  a  finite 
distance  from  O,  can  now  be  used  to  define  the  relation  between 
the  imaginary  and  infinite  points  introduced  into  each  system  of 
coordinates. 

There  appear,  then,  exceptional  points.  In  the  first  place,  we 
notice  that  the  tetracyclical  coi'irdinates  take  the  unallowed  values 
0:0:0:0  when  .r  -f  //'  -  -  <>,  f  -..-.  0.  That  is,  the  circle  points  at 
infinity  necessary  in  the  ('artesian  geometry  have  no  place  in  the 
tetracyclical  geometry.  Furthermore,  any  point  on  the  line  at 
infinity  t  =  0,  other  than  a  circle  point,  corresponds  to  the  real 
point  at  infinity  1:0:0:0  in  the  tetracyclical  coordinates. 

If  the  tetracyclical  coordinates  are  given,  the  ('artesian  coi'irdi- 
nates are  obtained  through  the  equations  .rt :  i/f :  t~  =  ./-, :  .r  :  ./•  .  These 
('([nations  will  determine  a  single  point  on  the  ('artesian  plane 
unless  .r  =  .r  =  a*  =  0.  In  this  case  /  —  0  and  the  ratio  ./•://  is 
indeterminate.  That  is,  the  real  point  at  infinity  in  tetracyclical 
coi'irdinates  corresponds  to  the  entire  line  at  infinity  in  ('artesian 
coi'irdinates.  Any  other  point  on  the  tetracyclical  locus  at  infinity 
.r  =  0  has  coordinates  of  the  form  j-  :  1  :  ±  /  :  0,  and  no  Cartesian 
coi'irdinates  can  be  found  corresponding  to  these  values. 

Hence,  ///  CftrtcKinn  cnf>r<JiH<ife8  n-t>  ////»/  <•<  rhnn  puintat^  (/»'  '•//•'•/»• 
points  nt  infinity^  icJu'ch  <l'>  ti»t  I'.rfxf  in  tffrrti't/ctn'tit  t'n<>rtlhi<ttt'>t<  <ni<l 
in  ti't  nii'//i'l  n'lil  I'niiril  i  iinti's  irr  find  t'crtrtln  jwintx,  t/H'  i)n<i</in<ii't(  points 
lit  infinity,  u'Jiii'h  iln  tint  c./-/.s7  ///  tin1  ( '<i rti-xi/i n  I'nUnlinntex.  11V  ii/*» 
fnul  tfnit  tin'  ri'iiJ  }>»int  nt  Iitti/tif//  in  fit  nit-//, •//,-,//  i'in'i'rdinatt'x  I'or/'i1- 
,sy/'///i/x  f/>  iln'  I'ntit'i'  lini1  nt  n/titt/f//  in  (ftit'f fit/tin  t'nffi'ifiniitt'tt,  iin<l,  <'<>/i- 
ViTXt'lif,  t/i'it  mi //  }><>///f  lit  inlinitil  in  ('iirtixnin  I'ui'i/'iltintti'K  ('ni'n'nimndit 
in  tin1  ri'iil  paint  of  infiniti/  in  tt'trttt'ifrlii'til  '•<"'''/-, li/nit,x.  \\  ith  these 
exceptions  the  relation  between  the  coordinates  is  one  to  one. 

The  exceptional  cases  bear  out  the  statements  in  s^:>>  and  I  as 
to  the  artificial  nature  of  the  conventions  as  to  imaginary  points 
and  points  at  infinity.  Since  the  ('artesian  coi'irdinates  are  more 
common,  there  is  some  danger  ot  thinking  that  the  conventions- 
there  made  are  in  some  way  essential.  The  discussion  of  this  ic\t 
shows,  however,  that  the  tetracyclical  conventions  ma\  be  made 
independently  of  the  Cartesian  ones,  ami  t  he  <_;vomet  r\  thus  deduced 
is  equally  as  valid  as  the  ('artesian.  As  lon<_r  as  either  set  ot 
coi'irdinates  is  used  by  itself,  the  difference  in  the  coii\entions  is 


144  TWO-DIMENSIONAL  GEOMETRY 

unnotieeable.  It  is  only  when  we  wish  to  pass  from  one  set  of 
coordinates  to  the  other  that  we  need  to  consider  this  difference. 
61.  Orthogonal  circles.  Consider  two  proper  circles  with  real 
centers  Cit  and  (\  and  real  radii  r(i  and  r,,  intersecting  in  a  real 
point  /'.  Then,  if  (  /'„,  /',,  )  is  the  angle  between  the  radii  <'„/'  and 
C,  /'.  and  '/  is  the  length  of  the  line  (',/',,,  we  have,  from  trigonometry, 


lint  the  angle  between  the  circles  is  either  equal  or  supple- 
mentary to  the  angle  between  their  radii.  Hence,  if  we  call  6  the 
anle  between  the  circles  we  have 


If  the  equations  of  the  two  circles  are 

«.'i+V,+  V,+  V4=°  (1) 

respectively,  the  formula  for  the  angle  may  be  reduced  by  (3),  §  o9, 
and  (4 ),  ^  -V,',  to  the  form 

_  -2  „  I  +  ,/  /,  4.  f,  I  -  -2  a  h 
cos0  =  ±    -  '  4=^ 

or,  more  compactl}', 

eos0  =  ±    ;    ?>("'M       ,  (3) 

\  17  (//)\  »;(/') 

where  7;  (  '/,  A)  is  the  polar  of  ?;(''). 

This  formula,  which  has  been  obtained  for  two  real  proper  circles 
intersecting  in   a   real  point,  is  now  taken   as  the  iliihiifion   of  the 

•~  1  . 

angle  between  any  two  circles  of  any  tvpes  whose  equations  are 
given  by  (1  )  and  (  '2  ).  We  leave  it  for  the  reader  to  show  that  if 
one  or  both  of  the  circles  is  ;i  real  straight  line,  the  definition 
agrees  with  the  usual  definition. 

The  condition   that    two  circles  should  be  orthogonal   is  then 

?/(",  />)=<>.  (4) 

If  the  circle  (1  )   is  ;i  special  circle,  the  coordinates  of  its  center 
have  been  shown  to  be  —  '2  </   :  <i.,:  <i , :  —  '2  n  .  and  equation  (  4  )  is  t he 


TKTKACVCMCAL  CO<  JKDIXATKS  It:, 

condition  that  this  center  should  lie  on  (_).  Hence'/  *i»  <'i<il  <•//•<•/>•, 
whether  <t  j">int  <-irff>'  <>r  <i  x/n'rinl  utriii'/ht  line.  in  "rf/i»</»ti<il  /<- 
(mother  circle  when  <nul  "/>///  ii'/ien  the  cfnti-r  <//'  tin'  xin'ri<i/  <•>/•••/<•  li>  * 
i>n  the  other  >•//•<•/«'. 

\Ve  might  equally  well  sav  that  a  special  circle  makes  anv  angle 
with  a  circle  on  which  its  center  lies,  since  in  Midi  a  case  cos  #  in 
('•}  )  is  indeterminate. 

It  is  possible  in  an  inlinitv  of  WHYS  to  find  four  circles  which 
are  mutuall  orthoonal.  For  if 


is  any  circle,  the  circle 

V/vr,=  0  (u) 

mav  l)e  found  in  x'  ways  orthogonal  to  (  5),  since  the  ratios  /-,  have 
to  satistv  onl\'  one  linear  equation  of  the  form  (4).  ('itvles  (  .">  > 
and  (t!)  being  fixed,  the  circle 

5>V,-=0  (7) 

IIKIY  be  found  in  an  infinite  number  of  wa\'s  orthogonal  to  (•>)  and 
(fi),  since  the  ratios  »•.  ha\e  to  satisfv  oiil\'  t\\o  linear  ecjuations. 

I''iiiall\',  1  he  circle 

V.,,-,^0 

may  be  found  orthogonal  to  (•">),  <  i|  >,  and  (7)  !>Y  solving  three 
linear  equations  for  e.. 

It    is  geometrically   evident    that    at    least    one  of    these    circles   is 
imaginary. 

EXERCISES 


orthogonal   and    find    a    fourth    circle   orthogonal    to   them. 

5.    I'rovr  that  r    --.  0,  ./•,  ..  -  0.  ./•      =  0  are   iniituall\    ort  lio^nnal.     ''an  a 
fourth  cin-lr  In-  found  orthogonal  to  them  '.'     K.\]ilain. 


14li  T\\<>    D1MKNSIONAL  (JKOMETKV 

6.  Find  all  circles  orthogonal  to  the  circle  at  infinity  .r  =  0. 

7.  I  nid   the   t'(|iiat  ions  of  all   circles  orthogonal   to  the   point  circle 
.,-,  -=  0.     lion-  do  they  lie  in  the  plane  '.' 

8.  Find   the  equations  of  all   circles   orthogonal    to   the   real    proper 


9.   Show  that  all  circles  whose  coefficients  <i  (  satisfy  a  linear  equation 

,•  ,/   -)_,.//   4-  f  a   4-  r  n   =0 

1     1     '        -1     -I     '        :i     3     '        44 

are  in  general  orthogonal  to  a  fixed  circle  and  find  that  circle. 
62.  Pencils  of  circles.    Consider  two  circles 

Vi+  'Vr'J+  'V':!  +  'V'4  =  °'  0  ) 

I  .r  4-  ?>  ./'  -f  ft  .r  -f  I  .r  =  0.  (  2  ) 

1      1  2     3    '         :i     3    '          44 

Witli    reference    to    them   we    shall    prove    first   the    following 
theorem  : 


I.   Ami  tn'"  fin'/!'*  hlti'l'SCCt  in  tiro  <tm]  <i>////  f/rn  jmftifx.     T/U'M  ft 
niiii/  /»'  nitinciili'nt,  in  it'hifh  cnxi'  tf/c  n  /'<•/>'*  <tr>'  xiiitl  to  /"'  t<i»</t'tit. 

To  prove  tins  we  note  that  if  equations  (1)  and  (2)  are  inde- 
pendent, at  least  one  of  the  determinants,  "/',—  "/',-.  must  be  different 
from  /.ero.  Hence  we  can  solve  for  one  pair  of  variables,  .r.  and  .r;. 
in  terms  of  the  other  two.  For  example,  we  mav  find  from  (  1  )  and 
(  '2  )  ./-j  =  f}.r,  +  ''.,-'',.  J'.,=  ''...''.,  -f-  '',-'",•  1  '  these  values  are  substituted 
in  the  fundamental  relation  «(./•)--  0,  there  results  a  (juuilratie 
cijiiation  in  ./•  and  ./4.  This  determines  t\\'o  values  of  ./-.  ;  :  ./•  .  and 
from  each  of  these  the  ratios  .r{:  .i\,  are  determined.  This  proves 
the  theorem. 

It  is  ''videiit  that  the  circle  points  at  infmitv  which  are  intro- 
duced as  a  convenient  fiction  in  ('artesian  <_;vometrv  do  not  appear 
here.  In  ('artesian  ovometrv  it  is  found  that  there  are  alwavs  two 
sets  of  coordinates  which  satisfv  the  equation  of  anv  circle,  and  we 
are  consequently  led  to  declare  that  all  circles  pass  through  the 
same  two  imaginary  points  at  inlinitv.  I>v  the  use  of  tetracvclical 
coordinates  there  are  no  two  points  at  inlinitv  common  to  all 
circles.  In  fact  the  circle  <\  >  meets  the  locus  at  inlinitv  ./•  -••-  (>  in 
the  t  wo  points  —  '/.,  T  ,i  /  ;  ,<  \  j  in  \  0.  which  are  not  the  same  for 
all  circles. 


TKTKACVCLICAL  (  '<  >f>Rl>INATKS  1  17 

Theorem  I  holds  ot  course  lor  the  ease  in  \vhicli  the  chvlo  are 
straight  lines,  one  ot  the  points  ui  intersection  bring  alwavs  the  real 
point  at  infinity.  Two  straight  lines  which  are  tangent  at  the  real 
point  at  inlinitv  are  parallel  lines  in  the  ('artesian  geometry. 

Consider  now  the  equation 


where  X  is  an  arbitrary  parameter.  For  any  value  of  X  (  -\  )  defines 
a  circle  which  passes  through  the  points  eoninion  to  (  1  >  and  (  '1  ) 
and  intersects  (1)  and  (-)  in  no  other  point.  The  totality  of  the 
circles  corresponding  to  all  values  of  X  forms  a  //»•/*••//  <<;'«•//•/•/,  •»•. 

If  (1  )  and  ('!)  are  real  circles,  the  pencil  ('•))  mav   lie  of  one  of 
the  following  tvpes  : 

(1  )  proper  circles  intersecting  in  the  same  two  real  points: 

('!)  proper  circles  intersecting  in  the  same  two  imaginary  points  : 

(  o  )  proper  envies  tangent  in  the  same  point  ; 

(4)  proper  concentric  circles; 

(  .")  )  a  pencil  of  intersecting  straight   lines; 

(ti)  a  pencil  of  parallel  straight   lines. 

II.  In  ///(//  I'l'ic-tl  ,,f  <-ir<-lfs  tlnTC  t*  "//<•  oii'l  «/i///  "/if  atnit'i/Jit  hiti', 
u/t/^xft  tin1  jH'/f/l  i-'iitxixfx  >'nttr<-lti  <>f  tstruit/Jtt  li/n-x. 

The  condition  that  ('•>)  should  represent  a  straight   line  is 


which  determines  one  and  onlv  one  value  of  X  unices  lioth  <t  and 
/'  are  y.ero.  In  the  latter  case  all  circles  defined  by  (  :>  )  ;"'*'  straight 
lines.  This  proves  the  theorem. 

'1  he  straight    line  ol    the  pencil   is  called   the   /•</<//••<//  tij'tx  ot    anv 
t\S'o  ein-les  ot    the  pellcll.      Its  eipiatloli    is 


This  is  a  special  line  when 

("./'    -  "'.,)"  4- (  "'-  "')"•     "• 


14S  TWO-DIMENSIONAL  GEOMETRY 

and   tin-  equations  (1),  ('2),  and    (-\)  represent   concentric  circles, 
and  the  radical  axis  is  the  line  at   infinity  .^=0. 

In   all  other  eases  the  radical  axis  of  two  real   circles  is  a  real 

straiht    line. 


///.  //;  'i  n  ii  i»-)t>-i/  of  i-irclex  (here  are  two  anil  on///  ttm  (ixnr  r>r 
i  i/i't'itictrii  )  Kfifi-ial  <•//•<•/<  '.v,  unless  the  pencil  nunaixttx  entirely  of  special 
clri'l,  *. 

l'>\    £  ."><»  the  condition  that  (•})  should  be  a  special  circle  is 

il  («  4-  X/')=  (\ 
or  ?;  (<>')  +'2\i)  (a,  /<  )  +  X-»;  (  '•  )  =  0. 


This  etjiiation  deti-nnines  two  distinct  or  equal  values  of  X 
unless  it  is  identically  satisfied.  Hence  the  theorem  is  proved. 

If  the  pencil  is  defined  by  two  real  proper  circles,  the  special 
circles  are  point  circles,  since  by  II  there  is  only  one  straight  line 
in  the  pencil  and  that  is  real  and  nonspecial.  It  is  not  difficult  to 
show  that  if  the  circles  of  the  pencil  intersect  in  real  points,  the 
special  circles  have  imaginary  centers  ;  if  the  circles  of  the  pencil 
intersect  iii  imaginary  points,  the  special  circles  have  real  centers; 
and  if  the  circles  of  the  pencil  are  tangent,  the  centers  of  the  special 
circles  coincide  at  the  point  of  tangenev. 

IV.  A  I'lnit'  arthof/iiiiid  f'.i  tn'n  <•//•<•/(>•  <>f  a  pem-il  /*  vrthoyuHid  to  all 

<v/v/rx  (•/  the  />en<'tl. 

Let  V'>r=  0  be  orthogonal  to  (1)  and  (_!).     Then 

7/(r,    rt)=0,      7>(>,   /0=  0; 

?;(<•,  n  +  X/-  )  =  7;  (<-,  ^~)4-X?/(c,  /-)=() 


lor  all  values  of  X.     This  proves  the  theorem. 

Il  lollows  from  this  and  v<  «il  that  a  circle  orthogonal  to  all 
eirelrs  o|  ;t  pt-ncil  passes  through  the  centers  of  the  special  circles 
ot  the  pencil,  and,  con  verselv,  a  circle  through  the  centers  of 
the  special  circles  is  orthogonal  to  all  circles  of  the  pencil.  If  the 
pencil  has  onlv  one  special  circle,  the  orthogonal  circles  can  be 
determined  as  circles  which  pass  through  the  center  of  the  special 
circle  and  are  ortliogonal  to  one  other  circle  of  the  pencil,  sav  the 
radical  axis. 


TETRACVPLICAL  COORDINATES         14',» 

These  considerations  lead  to  tin-  following  theorem  : 
V.  F<»r  nn if  jn'iK-11  <>f  i-in-li*  ///»•/•»•  f.rixtx  anotht't'  j»-n<-i(  m<r/i  t/t<it  n// 
cirdtK  of  citltiT  pencil  tt/'i'  orthoijonnl  to  dl!  nt'flfx  <>1  thf  <////»'/•,  <nt<l 
/mi/  I'in'lr  icJiich  /.s  urtJuxjoHdl  to  nil  <v'/vA'.v  '//'<///<•  JH'/K-/'/  fiflun«/n  t»  tJif 
othiT.  The  points  CH//I ///"/i  (<>  th.'  f/i'i'lcx  i >f  OKI'  jit'Ki'il  nrc  tlif  rt'/ttfrx 
of  th>'  ttfx'ct'nl  rirch'fi  of  tin-  of/it1/'. 

Fig.  47  shows  sm-li  mutually  orthogonal  pencils. 


EXERCISES 

1.  Show  that  two  real  circles  intersect    in   two  real  distinct    points, 
are  tangent,  or  intersect  in  two  conjugate  imaginary  points  according  as 

[TK'sW-^M^iO. 

2.  Show  that  the  point  circles  in  a  pencil  of  real  circles  have  real  and 
distinct,  conjugate  imaginary,  or  coincident   centers,  according  as   the 
circles  of  the  pencil  intersect  in  conjugate  imaginary,  real  and  distinct, 
or  coincident   points.     In  the  last  case  show  that  the  centers  of  the  point 
circles  coincide  with  the  point  of  tangencv  of  the  circles  of  the  pencil. 

3.  Show  that  circles  which  intellect  in  t  he  same  t  wo  j  points  at  infinity 
are  concent  ric. 

4.  I'rove  that  the  radical   axis  of  a   pencil  of  circles   passes  through 
the  centers  of  the  circles  of  the  orthogonal  pencil. 

5.  Prove  that  the  radical   axes  of   three  circles   not    belonging  to  the 
same  pencil  meet  in  a   point . 

G.  Take  V,,^^  0,  V /,,.,•,..<>,  V,.,..  =  0,  any  three  circles  not  be- 
longing to  the  same  pencil,  and  show  thai  ^  t  </,  -f-  A/'-- r  /*'',  i  •'',--" 
defines  a  two-dimensional  extent  of  circles  (a  rirrli-  <-,,/,,/,!,  .,•  \  eon>i^tiiiLr 
of  circles  ort  hogonal  ton  fixed  circle.  I  >iseuss  t  he  nnml  ier  a  nd  jio-^it  ion 
of  t  he  point  circles,  the  st  raigl it  line<,  and  t  he  special  lines  of  a  complex. 

7.  Show  that  the  totality  of  straight  lines  form  a  complex.  To  what 
circle  are  t  he\-  ort  IK  >gonal  '.' 

S.    Show  that  circles  common  to  two  complexes  form  a   pencil. 


1")0  TWO-DIMENSIONAL  GEOMETRY 

63.  The  general  tetracyclical  coordinates.  Let  us  take  as  circles 
of  reference  any  four  circles  not  intersecting  in  the  same  point 
and  the  equations  of  which,  in  the  special  tetracyclical  coordinates 
thus  far  used,  are 


and   let    us  plac 


P  A^  = 


Since  the  four  circles  do  not  meet  in  a  point  their  equations 
cannot  he  satisfied  by  the  same  values  of  .r,  and  therefore  the 
determinant  of  the  coefficients  in  (1)  does  not  vanish.  Therefore 
the  equations  can  be  solved  for  .r.  with  the  result 


where  A.  is  the  eofactor  of  a,  in  the  determinant  of  the,  coefficients 
of  (1  ),  B  the  eofactor  of  /^,,  etc. 

The  relation  between  the  ratios  ./•  :  .r,  :./'.,:  ./•  and  A'{  :  A,  :  A',  :  .\"4 
is  therefore  one  to  one,  and  the  latter  ratios  niav  be  taken  as  the 
coi'irdinates  of  anv  point.  These  are  the  most  general  tetracyclical 
coi'irdinates. 

A  ^eonu-tric  meaning  may  be  given  to  these  coordinates  as 
follows  : 

It   the  circle  with  the  ('artesian  equation 

'/(.'•-+//-)+  IJT  +  '•//  +(7  =  0 

is  a  real  proper  ciivlc,  and  the  point  /'(./',  //  )  is  a  real  point  outside 
ot  it.  then  the  expression 


i-  proportional  to  the  fnnrt-r  of  /'  with  respect   to  the  circle  :   that   is, 
to  !  he  length  of  the  square  of  the  tangent   from  /'  t,o  t.he  circl'1.     It 


TKTKACVCLICAL  COORDINATES  1-',1 

7'  is  a  real  point  inside  the  circle,  the  power  may  lie  defined  as  the 
product    <il    the  lengths   of  the   segments  of  any   chord   through  /'. 

Also,  if 

/,,.  +  ,.v  +  ,/  =  o 

is  a  real  straight  line,  the  expressinn 


is  proportional  to  the  length   of  the  perpendicular  from   any  real 
point  to  the  line. 

l>y  yirtne  of  £  til)  these  relations  hold  for  a  linear  equation  in 
tetracyclical  coordinates.  Of  course  if  the  [mints,  circles,  or  lines 
involved  are  imaginary,  the  phraseology  is  largely  a  matter  of 
definition.  We  may  say,  then  : 

Thf  ntoxt  general  ti'tracyi'lii'al  nx'irdinatex  »f  a  point  cnnsixt  of  (/«• 
rutiox  of  four  quantities  each  of  which  ix  equal  to  <i  constant  tiinf*  tin' 
pou'i'r  of  //(,>  point  irith  reference  to  it  i-lffle  of  reference,  o/\  In  raxe 
tJie  circle  of  reference  /*  a  xtrn'njht  line,  to  a  constant  fi//ti'x  the  leiujth 
of  thf  perpendicular  from  tin'  point  to  ?//>•  line.* 

I>y  means  of  (1)  the  fundamental  relation  o>  (./•)=  0  goes  o\er 
into  the  new  fundamental  relation 


and  the  polar  equation  w(.r,  //)  =  0  becomes 


here  the  determinant  \nik\  does  not  vanish. 
The  real  point  at  infinity  has  now  the  coordinates  X^  :  A'^  :  A'{  :  A'4 
=  0:^/3^7:8,    and     hence    liy    a    proper    choice    of    the    circles    of 
reterciice    may    he    given    any    desired    coordinates.     The    locus    at 
intinity   has   the   equation 


*  Si  line  aut  tmrs  prefer  tu  ilctinc  t  lie  cmirilinate  as  \  lie  c|iint  ieiii  nf  I  he  |n  >  \\vi-  >  'f  I  he 

pnilit    iliviileil    l>y    the    railillS.   silll'C    this   i  jlli  it  it-Ill    LT'H'S  HVfl'   illl"    I  wire    the    lenuth   i'l' 

the  [.erpenilieiilar  1'niiii  the  ]Miint  tu    a  straight    lim    \\lieii  the   nuliu.-  nl'  the  rin-li1 

heeolnes  illlillite.  This  tlelinit  ii  m  fails  if  the  ril'ele  uf  reference  i>  a  |"'int  rirele 
when  the  e'in-e>]iMiiilillu  i''M)l-i|ili:ite  is  the  >.|liare  i  if  the  diMaiiee  nf  the  ptiint  fl'iilll 
the  center  uf  the  eilvle.  Siliee  tile  1-1  ilistulll  \\hieli  may  lliultiply  eaell  i'i  ii'iI'ililUlti1  is 

arliitrar\.  \\  e  prefer  the  iletinit  inn  in  the  text. 


l.VJ  TWO   DIMENSIONAL  GEOMETRY 

A  circle  \vith  the  equation 

V'l+V2+V3+a4-r4=° 

luis  in  tin.1  iH'\v  coordinates  the  equation 


By   virtue   of   these   relations  the   conditioii   for  a  special   eirele 
j,t—()   becomes   a  new   relation 


and  the  eundition  ?/  (_a,  /<)  =  0  for  orthogonal  circles  becomes 

H(,l,  70  =  2/,,AA=<>.  (7) 

The  form  H  (  .  I  )  may  be  computed  directly  from  11  (A')  as  follows  : 
l!v   formulas  (4)  and  ('2),  §  .r)S,  the  et|uation  of  a  point  circle 

with  the  center  K   is 

H(A",  }')=  0. 

Hence,  if  A^+A^+A^+A^X^  0 

is  a  point  circle,  we  must  have 

^,-=«,-ii'i+««^a+«air8+fl«>V  C8) 

These  (Mjuations  can  be  solved  for  )',  since  the  determinant  '  ^.^  i 
docs  not  vanish.  But  T  bein^  the  coiirdinates  of  a  point  must  sat- 
isfy the  fundamental  relation  (3).  Substituting,  we  obtain  a  rela- 
tion between  the  .I's  to  be  satisfied  by  any  point  circle.  This  can 
b»-  nothing  else  than  the  condition 

H  (-0=0. 
By  virtue  of  (*)  we  have,  accordingly, 

II  00=  MK  )'). 

But  (*)  can  be  written  <r.\.  - 

<  }\ 

' 


Hence  we  have  H  ('  —  )  _  ATI  (  >').  (9) 


TKTRAC  YCLK'AL  ('<  >(  >KDI  NATES  1  .',;} 

Also  the  form  fl(.V)  may  he  computed  from  the  form  II  (A)  as 
follows:  If  A  is  a  point  circle,  equation  (  7  )  expresses  the  condition 
that  the  center  of  A  should  lie  on  a  circle  l>.  But  if  *\\  are  the 
coordinates,  ot  the  center  ot  A,  this  condition  is 


Hence,  hv  comparison  with  (7), 

M^A,,.  !,  +  /-„.  L-H;1.l.  +  /'l4./<.  (10) 

Since./  is  a  point  circle  its  coefficients  Ai  satisfy  (  •  '»  ).    Therefore. 

if  equations  (1<>)  an-  solved  for  J,  and  the  result  suhstituted   in 

(  ii  ),  we  have  a  relation  satisfied  hy  the  coordinates  of  any  point. 
This  can  only  be                      ^     \    —  o 

By  virtue  of  (10)  we  have,  accordingly, 


But  (In)  can  he  written        <r.V  = 

rAt 

Hence  we  have  ft  (  —  U-  A"  II  (  ,1).  (11) 

\cAJ 

64.  Orthogonal  coordinates.  Particular  interest  attaches  to  the  cast- 
in  which  the  four  circles  of  reference  are  mutually  orthogonal.  If 
the  circle  .\'.=  0  is  orthogonal  to  the  circle  A\  —  0,  \\-e  have,  from  (  7  ). 
vj  <»:;,  /,.(  .—  il.  Therefore,  for  an  orthogonal  svsteni  of  coordinates. 
we  have 

H  (.  I  )=/-,,/,-+  /",.r;+^-  ':=  +  /'-4-<;. 

Mipiat  ii  ins  (  1  o  ),  ;<  fl:},  ^ive 

p.\\=  Av.l,., 
whence  the  tundamental  relation  tor  the  point  ooinxlinates  is 


\\'ithoiit  ehangijiLj  the  coi'irdinate  circles  it   is  ohviously  pos>il»le 
to  change  the  coft'ticieiits  in  (  1  ).       M':!.  so  that  /.•,—  !.    Then  we  have 


l  TWO-DIMENSIONAL  GEOMETRY 

A  special  case  is  obtained  by  placing 


where  ./•_  are  the  special  coordinates  of  ^  ->7.  The  four  circles  of 
reference  are  a  real  circle  with  center  at  (>  and  radius  1,  two  per- 
pendicular straight  lines  through  o,  and  an  imaginary  circle  with 
center  at  <>  and  radius  /. 

65.  The  linear  transformation.  Let  ./•.  be  any  set  (special  or 
general)  of  tetracyclical  coordinates  where  &>(./•)—<)  is  the  fun- 
damental relation,  and  consider  the  transformation  defined  by  the 
equations 

f»'~  «,!*!+  V,+  <V-,+  <V> 

p4=s«al*i+<*&ra+va+«*f*> 

p*=  Vi+  <«+  Va+'VV 

p.r(  =  a   .r-f-  a   x  -f  <f   •/'  +  <t   .?*  , 

'         4  .|1       1      '  4-J      o      I  .,;;      ;;      I  .,4      4) 

when1  the  determinant  of  the  coet'licients  '  <t  a.:  does  not  vanish  and 
where  ./•'  satisfies  the  same  fundamental  relation  as  .r{. 

J>v  means  of  (1  )  any  point  ./;  is  transformed  into  a  point  ./,  and 
since  the  equations  can  be  solved  for  ./•_,  the  relation  between  a 
point  and  its  transformed  point  is  one  to  one. 

By  means  of  (1),  also,  any  circle 

a  ./•  -f-  <'  •''  +  ''  •''  +  "  •''  =  (^ 

i    i   '       2   i;   '       3   a   '       -14 

is  transformed  into  the  circle 


where  P'<[-~   -\t\«\\-  '\;-'(ti  +  -{,••,«:',  +  ^i^.1- 

Now,  if  //_  is  a  fixed  point.  j\  a  variable  point,  and  //'  and   ./•[  the 
transformed  points  respectively,  the  equation 

w(./-,  #)=  0 
is  t  ransti  irnied  into  1  lie  equal  i<  >n 

ro  (  r',  //')  =  0, 
since  the  t-quat  n  iii  f<>  (  ./  )  =  0  is  transformed  into  ro  (./•')=  0. 


TETKACVCLICAL  COORDINATES  I-").") 

That  is,  /<//  tin'  tranxtnnnatinn  (1  )  special  cirrlt'x  <tr>'  tr<i>ixf»r>iit'</ 
info  xpccittl  i-irc/i-ft,  tin-  <•>'//{<'/•  of  each  special  circle  ocini/  transformed 
into  tin'  center  <>t  tin'  transformed  circle. 

It  follows  troiu  the  above  that  nonfpecial  circles  are  transformed 
into  nonxpecial  cir<-/fx,  for  if  a  nonspecial  circle  were  transformed 
into  a  special  circle,  the  inverse  transformation  would  transform  a 
special  circle  into  a  nonspecial  circle,  and  since  the  inverse  trans- 
forination  is  also  of  the  form  (  1  ),  this  is  impossible. 

We  mav  accordingly  infer  that  hv  the  transformation  (1)  the 
equation  ?/(>)=  0  is  transformed  into  itself. 

\Ve  may  distinguish  between  two  main  classes  of  transformations 
of  the  form  (1)  according  as  the  real  point  at  intinitv  is  invariant 
or  not.  The  truth  of  the  following  theorem  is  evident  : 

//  a  linciir  transformation  leaves,  tin'  mil  //"////  ,it  infinity  invariant, 
eren/  iftrat'i/ht  l/>ii'  is  tnt unformed  into  <i  straight  ////»•  <nnl  >•>',•/•>/  j>r<>j»  /• 
rifflf  into  a  i>r»p<'r  <•/'/•<•/>'.  It'  <i  lim'nr  tritnxfnrwrtfinn  tniH#f<>r)n>t  f/f 
n'/if  point  <it  infinity  into  n  point  < >  cnnl  transforms  a  point  (>'  into 
tin'  fi'iil  point  nt  /'/itiii/t//,  nni/  xtntii/lit  Hin1  ix  transformed  info  a  <-lr<-1<- 
throiii/h  < >,  <t ml  <im/  /•//•'•/»•  thnnujh  < >'  i*  transformed  into  a  straiylit  ////»•. 

Since,  as  we  have  seen,  the  c(|iiat ion  ?/('')=  "  is  transformed  into 
itself,  \\'c  mav  write  */ (''')  =  /">/(")<  the  value  of  /•  di'pt'liding  on 
the  factor /3  in  (1).  \\'ith  the  same  factor  we  have  ?/ (//)=/.•?/(/.) 
and  ?/("'.  /'')=/-?/  (<r,  //).  Hence  by  ( :> ),  £  '"'I,  the  angle  between 
t\\o  circles  is  equal  to  the  angle  between  the  two  transformed 
circles.  Tin1  liin'itr  transformation  i*  tJifrefnre  conformaL 

66.  The  metrical  transformation.  We  shall  pro\e  lirst  that  n>i;i 
transformation  of  tin'  metrical  i/roup  c<m  o,-  ejrprt'*>ti'J  ax  a  Innin- 
transformation  of  tt'tracrfdieal  coiinlinatest. 

We  have  seen  in  ^  4.)  that  a  transformation  of  the  metrical  group 
is  a  linear  transformation  of  the  ('artesian  coordinates  ./•  and  // 
together  with  the  condition  ( ./•'"+  //'" )  ---  /r  (./•''+  //'*').  It  fol!o\\s  from 
this  that  the  transformation  can  be  expressed  as  a  linear  transfor- 
mation of  the  special  coordinates  of  sj  ;~>7.  But  the  general  tetra- 
cyclical  coi'irdmates  are  linear  combinations  of  the  special  ones. 
Hence  t he  the* irein  is  [ in >ved. 

Since  a  metrical  transformation  transforms  straight  lines  into 
straight  lines,  it  must  leave  the  real  point  at  infinity  invariant. 


lofi  TWO    PIMKNSIONAL   CKOMKTKV 

(  'on  verse  1  \.  tin  i/  liih  <ir  ti'unxfornifltion  »f  tetrtii'UcUi'dl  cutifiUnntcs 
tchi''h  1<  'iti'fn  tin-  ri'iil  jmint  iff  infinity  ini'<tn<int  fx  n  tranxfonnntivn  <>f 
th>  i/ii'trfi-'il  ///•"///>.  This  may  lie  shown  ;ts  follows: 

If  thf  real  point  at  infinity  is  invariant,  the  locus  at  infinity  is 
transformed  into  itself,  since  it  is  a  special  circle  with  its  center  at 
the  real  point  at  intinitv.  Therefore  any  linear  transformation  of 

ovne  ral  tet  racvclical  coordinates  which  leaves  the  real  point  at  infinity 
^  .  i  * 

invariant  is  equivalent  to  a  transformation  of  the  special  coordinates 
of  vj  ~>7,  which  leaves  the  point  1  :  0  :  0  ;  (I  invariant  and  transforms 
the  locus  ./•  =  0  into  itself;  that  is,  to  a  transformation  of  the  form 


= 

Since  r!*+  ./f  -  stf  =  Jr  (./-;  4-  ./•;  -  ./v4), 

we  have,  for  the  coefficients,  the  conditions 

k2 
al^  a*i=  ai*+  n^=  au=  "3' 


Now  the  last   three  equations  of  (  1  )  are  equivalent  to  the  equa- 
tions in  Cartesian  coordinates 

./•'  ~  n    .>•  -\-  n    i/  -f-  a  ,  , 


and  the  conditions  imposed  on  the  coefficients  are  exactly  those 
necessary  to  make  this  a  metrical  transformation.  The  first  equa- 
tion in  (  1  )  is  a  consequence  of  the  last  three  equations  in  (  1  )  and 
the  condition  (  '1  ).  In  fact,  the  coefficients  a,,,  'i,,.  a.,,  and  't.,.,  may 
first  he  determined  to  satisfy  equations  (  •>  ),  the  coefficients  a,  and 
'r,(  may  lie  assumed  arliit  varilv,  and  the  coefficients  a  ,  n^.  ft  , 
and  n^  are  then  determined  liy  (  '•]  ).  This  jiroves  the  theorem. 

67.  Inversion.  T\\n  points  /'and  /''  are  ///'wx/-  witli  respect  to  a 
nonspecial  circle  <'  it  e\erv  circle  through  /'  and  /*'  is  urtho^onal 
to  ('.  I;i-oiu  this  it  follows  that  if  (_'  is  a  straight  line  two  inverse 


TETRACYt'LICAL  <  '<  ><  iKDIXATKS 


points  art-  symmetrical  with  respect  t<>  that  line  ;  that  is,  the  straight 
line  /'/''  is  perpendicular  to  < '  and  bisected  liy  it.  By  a  limit  procos 
it  is  natural  to  define  the  inverse  of  a  point  on  the  straight  line  (.' 
as  tin1  point  itself. 

If  ('  is  a  proper  circle  with  radius  /•  and  center  J  (J*'ig.  4*),  the 
inverse  of  ./  is  the  real  point  at  infinity,  since  the  circles  which 
pass  through  A  and  the  real  point  at  infinity  are  straight  lines 
perpendicular  to  <'.  If  /'  is  not  at  .1 
nor  on  (',  the  straight  line  /'/''  must 
pass  through  .1,  since  that  line  is  a 
circle  through  /'  and  J''  which  by  defi- 
nition must  he  orthogonal  to  ' '.  Take 
now  the  point  M  midway  between  /' 
and  /''  so  that 

AM---:  \  (.!/'  +  .//''), 

,         .   ,  KM..    18 

and   with    .'/   as   a    center    construct    a 

circle    through     /'    and     /''.     If     //    is    the     radius    of    this    circle. 

/,'=  l(M>'  -AT). 

I>v  squaring  the  last  two  equations  and  subtracting  one  from  the 

other,  we  have  ,  ,.-       /.-_    , /.      <  /,' 

.  i . M    —  / 1    —  ^  i  /    • _ i 1    . 

I>ut  the  condition  foi-  orthogonal  circles  gives 
/.''-+  r*—A.\r=  0. 

Hence  \\'e  have  as  the  condition  satisfied  by  two  inverse  points 
with  respect  to  a  circle  with  radius  /•  and  center  .1 

.I/'.  .l/>?=  r.  (  1  ) 

Conversely,  if  /'and  /''  arc  t\\'o  points  so  placed  that  the  line 
/'/''  passes  through  A  and  the  condition  (  1  )  is  sat  is  tied,  t  he  line  /'/'' 
and  the  circle  described  on  /'/''as  a  diameter  are  ea>ilv  [irovetl  tn 
be  orthogonal  to  ('.  Then  any  circle  through  /'  and  /''  is  orthogonal 
to  ('  by  theorem  IV.  ^  (I:!,  lleiice  /'  and  /''  arc  inverse  points. 

The  condition  (1)  shows  that  if  one  of  the  point<  /'and  /'  i> 

inside  of  the  circle,  tl ther  is  out>idc  of  it.  The  condition  holds 

also  for  the  point  .1,  since  if  .I/'  < ).  .I/''  s.  .  \\\  a  natural 
extension  of  the  definition  of  inverse  points,  condition  (  1  )  can  aUo 

1 

be  taken  to  hold  for  a  point  on  the  circle  <',  so  that  we  ma\  >av 
that  any  point  on  the  circle  ('  is  its  own  inverse. 


108 


TWO   DIMKNSloNAL  (JKOMKTIIV 


It  is  to  be  noticed  that  inverse  points  as  here  defined  are  also 
inverse  in  the  sen>e  of  ^  ~>-}>  if  the  circle  ('  is  a  proper  circle,  but 
the  definition  c/iven  in  this  section  is  wider  than  that  in  £  ;>o,  since 
it  holds  when  the  circle  becomes  a  straight  line. 

An  iivfi'ti'in  with  respect  to  a  nonspeeial  circle  ('  is  defined  as 
a  point  transformation  bv  which  each  point  of  the  plane  is  trans- 
formed into  its  inverse  point  with  respect  to  that  circle.  We  shall 
proceed  to  prove  that  <mi/  inr,'t'x/<»i  ''<in  /«•  ri'^rcKi'nti'd  li/  a  lin^nr 
tritn*i'<inn<tti»n  <•/  /•'/•'/'•//<•//<•»//  f»n'r</!>«ift'*.  It  is  first  of  all  to 
be  noticed  that  bv  an  inversion  each  point  of  the  circle  ('  is 
left  unchanged  bv  the  inversion.  This  condition  is  met  bv  the 

transformation  x^ 

p.i-.  —  A.,r  -f  ".  >  >'kJ\,  (-  ) 

^•hereN  <•,  .r,  —  <•  is  the  e(|iiation  of  ( '.  Now  let  ^  /»,..'', =  0  be  anv 
circle  through  j\  and  its  transformed  point  .r'.  Since  ^  ''',•'',•=  "  and 
V/vr'=  0,  we  have,  from  (  2), 

,,  f,  4-  „  /,  4-  „  /,  _f-  ,/  /,  =  i).  /;>) 

11  J    J  :',;)'        44 

If  V/',,/-,  =  o  is  orthogonal  to  ( ',  we  have 

,..i 

(4) 

and  therefore  if  (4)  is  satisfied  by  all  values  of /-!  which  satisfy  (•''>), 


It  remains  to  determine  X.  For  that  purpose  we  use  the  con- 
dition that  (i)  (./•)=  l(  and  w(r')=0,  and  lor  convenience  writiii'_r  A 
in  place  of  the  svmbol  "^  <'t.i'f.  \\'e  have 


But  CD  ( it  )—  co[     -}  and.  bv 
1 


II.-: 


+  •' 


TKT  K  A  < '  V( '  L  K '  A  L  ( '< )('  JK DIN  A  T  KS  1  -V. ) 

rr,          c  1  •y-v      ro)        A"  ^--v  K 

Therefore        &>  (.r,  a)  =  -  ^.r,     -  =  ^  2/V,  =  .,  -1, 

and,  from  (•">),  A.  =  —     <y(a)  = —  V  (,'')• 

We  have  consequently  built  up  the  transformation 


\vhich  is  an  inverse  transformation,  since  it  transforms  any  point  rt 
into  a  point  ./•'  such  that  anv  circle  through  r,  and  .r(  is  orthogonal 
to  f.  The  theorem  is  therefore  proved.  It  is  to  be  noticed  that  the 
transformation  is  completely  determined  when  the  circle  ('  is  known. 
68.  The  linear  group.  We  are  now  prepared  to  prove  the  fol- 
lowing proposition  : 

Any  linear  transformation  l>i/  ?/'///<•//  tJn'  r>'«l  point  «t  ///thtif//  i# 
invariant  <>r  ?'.*•'  transformed  into  n  paint  n<>t  <it  infinity  ix  tin'  pr«ihi<'t 
(if  <m  invention  <in<l  a  metrical  transformation, 

To  prove  this  let  T  be  a  transformation  of  the  form 

p.r'  -  a, ^  +  «,-«:»•,  +  ai3rs  +  arf\, 
l)v  means  of  which  the  relation  a>(af)—  0  is  transformed  into  itself. 

If  the  real  point  at  infinity  is  invariant,  the  transformation  is 
metrical  (§  <!tj).  If  the  real  point  at  infinity  is  transformed  into  a 
finite  point  J,  let  A  be  taken  as  the  center  of  a  circle  ("with  respect 
to  which  an  inversion  7  is  carried  out.  By  /the  point  A  goes  into 
the  real  point  at  infinity.  Hence  the  product  IT  leaves  the  point  at 
infmitv  invariant  and  is  therefore  a  metrical  transformation,  ('all 
it  .)/.  Then  IT—  !/• 

whence  7'=/-'.J/=y.lA 

\Ve  have  written  /"'  — /  because  an  inversion  repeated  gives  the 
identical  transformation,  and  hence  an  inversion  is  its  own  inverse. 

The  tet  racyclical  coordinates  are  adapted  to  the  study  ol  the 
properties  of  figures  which  are  not  altered  bv  this  group  of  linear 
transformations.  In  the  g<'ometr\'  of  these  properties  the  straight 
line  is  not  to  he  distinguished  from  a  circle,  since  anv  piunt  ol  the 
plane  mav  be  transl'ornieil  into  the  real  point  at  intinitv.  and  thereby 
anv  circle  mav  be  transformed  into  a  straight  line  and  vice  Versa. 
Anv  pencil  of  circles  mav  in  this  way  be  transformed  into  a  pencil 


IliO  TWO   DIMKNSIONAL  CKOMKTHV 

of  straight   lines  and  nianv  properties  of   pencils  of  circles  obtained 
from  the  more  evident   properties  of  pencils  of  straight  lines. 

The  distinction  between  special  and  nonspccial  circles  is,  how- 
ever, fundamental,  since  a  circle  ot  one  of  these  classes  is  trans- 
formed into  a  circle  ol  the  same  class. 

EXERCISES 

1.  Write  formulas  (C>\  ;?  ('(7.  for  the  special  coordinates  of  £  ."  and 
for  the  orthogonal  coordinates  of  j  (11. 

'2.  From  (Co,  §  C>7.  obtain  in  the  coordinates  of  ?  f>7  the  formulas  for 
inversion  on  the  circle  of  unit  radius  with  its  center  at  the  origin,  and 
cheek  bv  changing  to  ('artesian  coordinates. 

3.  Show    from    ((>),    £  C>7,   that     inversion    on    a    fundamental    circle 
of    a  svstem   of  orthogonal   coordinates    is   expressed    by  changing   the 
sign  of  the  corresponding  coordinate  and  leaving  the  other  coordinates 
unchanged. 

4.  Prove   that   a   plane   ligure   is   unchanged   bv    four   inversions   on 
four   orthogonal    circles. 

5.  Show  that   three  inversions  on  orthogonal  circles  have  the  same 
effect  as  an  inversion  on  a  fourth  circle  orthogonal  to  the  three. 

(>.  Prove  that  the  product  of  two  inversions  is  commutative  when 
and  only  when  they  take  place  with  reference  to  orthogonal  circles. 

7.  Show  that   the  product  of  two  inversions  on  two  straight   lines  is 
a  rotation  about  the  point  of  intersect ion>  of  the  two  lines. 

8.  \\\   Kx.  7  show  that  the   product    of  two  inversions  on  the  circles 
('    and  ('_,  can  be  replaced  by  the  product  of  the  inversions  on  two  cir- 
cles ''I  and  ' '.'  if  *"|  and   f '.',  pass  through  an   intersection  of  '^and   <' 
and  make  the  same  angle  with  each  other. 

'.(.    ('oii>ider  the  curve  defined  by  the  quadratic  equation 

v  „„,,-,,.,.  =  o. 

-*-^ 

Show  that  any  circle  or  straight  line  intersects  the  curve  in  four 
pomi^.  It  the  coordinates  are  the  special  coordinates  of  S  T»7.  classify 
the  curve  according  as  (1)  il  does  not  pass  through  the  real  point  at 
infinity,  i  '_'  >  \\  passes  once  through  the  real  point  at  infinity,  i  .">  i  it 
passes  twice  through  the  ]-eal  point  at  infinity.  <H>tain  the  ('artesian 
for  each  o|'  tlie  classes  and  note  the  ivl;itii>n  of  the  curve  to 
tin  circular  point-  at  infinity.  Note  that  the  above  classification  is 
inie-^ciil  ial  from  the  >tandpoint  of  the  linear  group  <if  tetracvdical 
t.ran>fi  irma  tions. 


TKTKACVCLK'AL  COORDINATES  Itil 

69.  Duals  of  tetracyclical  coordinates.  \>\  anticipating  a  little  of 
tin*  discussion  of  space  geometry,  to  he  j^ivrn  later,  \ve  mav  obtain 
duals  to  the  tetracyclical  coordinates.  The  student  to  whom  space 
^•eoinetrv  is  unknown  mav  post[ione  the  reading  of  this  section. 

If  we  interpret  the  ratios  j-  :  j-  :  s  :  j-  as  quadriplanar  point 
coordinates  in  space  of  three  dimensions,  then 

*>(,•'•>  ~V  (1) 

is  a  sui'face  of  second  order,  and  the  <jfeomet  rv  on  this  surface  is 
dualistic  with  the  j_;vometry  in  the  plane  obtained  bv  the  use  of 
tetracvclical  coordinates. 

The  linear  e<[iiation  ^^  r,-=  0  represents  the  plane  section  of 
the  surface  (1  ),  and  these  sections  are  the  duals  of  the  circles  in 
the  plane.  The  point  at  infinity  is  a  point  on  (1  )  not  necessarily 
geometrically  peculiar,  and  the  straight  lines  in  the  tetraeyelical 
plane  are  duals  to  the  plane  sections  of  (1  )  through  this  point. 

More  specifically  let  us  consider  the  speciali/.ed  coordinates  of 
£  .•>?  and  place  in  space  ./^  :  .r, :  .r( :  .r(=  z  :  .r  :  //  :  f,  the  usual  homoge- 
neous Cartesian  coordinates.  The  fundamental  equation  is  now 
the  equation  _r'+  //s_^==0j 

A\'hich.  in  space,  represents  an  elliptic  paraboloid.  We  have,  then, 
the  following  dualistic  properties: 

Tin-  rll'uitic  iHifabnlmd 

The  point  at   infinity  on  <>Z. 

Any  plane  seet  ion. 

A  n  eilipt  ie  seet  i<  in  made  l>v  a 
plane  n<  it  pai'allel  to  <  >/. . 

A  parabolic  section  made  by  a 
plane  parallel  to  <>'/.. 

A  section  made  bv  a  tangent 
plane. 

A  section  mad'1  hv  a  tangent 
plane  nut  jiai'allcl  ID  <  > /. . 


The  real  point  at  inlinit  y. 


Iti-J  TWO   DI.MKNSloNAL  (JKOMETUY 

Again,  it  \\e  have  tetracyclical  coordinates  for  which  the  funda- 
nu-iitnl  equation  is  rf  4-  r.r  4-  r;  -  j-43  =  0, 

\\liich  ran  lie  obtained  from  tin-  special  orthogonal  system  given 
in  jjtit  liv  mnltiplving  .r(  liy  /,  the  geometry  obtained  thereby  is 
dualistir  with  the  c_;'e<  unet  rv  on  the  surface  of  the  sphere 

./•-  4-  >/-  +  2-  =  1  . 

Iii  this  ease  the  tetraeyelieal  point  at  infinity  is  dualistir  to  the 
point  A",  where  the  sphere  is  cut  by  (>Z.  Circles  on  the  tetraeyelieal 
plane  are  dualist  ie  to  circles  on  the  sphere,  the  straight  lines  on 
the  plane  corresponding  to  circles  through  the  point  X  on  the 
sphere.  This  brings  into  clear  light  the  absolute  equivalence  of  a 
straight  line  and  circle  by  the  use  of  tetracvelical  coordinates.  In 
fact,  the  plane  ireometrv  on  the  tetracvclical  plane  is  the  stereo- 

.  «/  1 

graphic  projection  of  the  spherical  geometrv. 
To  see  this  take  the  sphere  whose  equation  is 

./•-+//-+r=l, 

and  let  A"  (0.  <>.  1  )  be  a  fixed  point  on  it  and  /'  (  £,  7;,  £  )  any  point 
on  it.  The  equation  of  the  straight  line  XI'  is 


and    this    line    intersects    the    plane    ^  =  0    in    a   point    Q   with    the 

coordinates  t  7, 

./•  =         -•>          //  =  • 

i  -  r          i  -  ? 

From    these    equations   and    the   equation    £~  +  if  +  £""  =  1  ,   which 
expresses  the  fact    that    /'  is  on  the  sphere,  we  mav  compute 

fc  -•'•  -//  r-'^+jll    ], 

.'••  f  if  4-1  7/  "  /"  +  //•  +  1  ~  J--  +  >r  4-  1  ' 


p.r   =j-~+  t/~—~[, 


p.l-       -  //. 
P-'\-  x~+ 


TKTKACYCUCAL  COORDINATES  li;;} 

Xo\v,  on  tht!  one  hand,  ./•  :./•,:  ./^  :  .r(  arc  homogeneous  ('artesian 
coordinates  of  a  point  on  the  sphere,  and,  on  t  he  ot  her  hand,  t  hrv  are 
tetracyclical  coordinates  of  a  point  on  the  plane,  heiiiLj  connected 
with  the  speeiali/.ed  coordinates  of  £  f>7  hy  the  c(juati(»ns 

pj-t  =  ./•;  -  j-'t,       p.r.,  =  -2  ./•;.       p.r.  =  -2  j-'3,       p.rt  -=  ./'4  +  r4', 

where  j-[  :  r.',  :./•.':  ./-4'  are  the  special  courdinates. 

From  this  relation  we  may  read  off  the  following  dnalistie 
properties  : 


Anv  point  of  the  plane.  Any  point  on  the  sphere. 

The  point  at  iniinity.  The  point  \. 

Any  eirele.  A  circle  (anv  plane  section). 

A  straight  line.  A  circle  through  \. 

A  special  circle.  A    section    made    hy    a    tangent 

plane. 
A  j)oint  circle.  A    section    made    liv   a    tangent 

])laiie  not  passing  through  .V. 
The  center  of  a  point  circle.  The    point    of    tan^eiicy    of   the 

tangent  jilane. 
A  special  straight  line.  A        tangent        plane       jiassini: 

through  A'. 

The  center  of  a  special  straight  A  point   on   the  plane  :.  =  1   not 

line.  coincident  with  A'. 

The  special  line  at  intinity.  The  section  made  hy  the   plane 

,-.-  =  1  (a  tangent   plane  i. 
Parallel  lines.  Circles    tangent     to    each    other 

at  .V. 


rilAPTKU  X 


A  SPECIAL  SYSTEM  OF  COORDINATES 

70.  The  coordinate  system.  Kadi  of  the  two  coordinates  x  and  // 
in  a  Cartesian  system  is  ot  the  type  described  in  £  7  for  the  coordi- 
nate of  a  point  on  a  line.  An  interesting  example  of  a  more  general 
type  of  coordinates  may  he  obtained  by  taking  each  of  the  coordi- 
nates in  the  manner  described  in  £  S.  \Ve  shall  develop  a  little  of 
the  geometry  obtained.  The  results  will  be  of  importance  chiefly  as 
showing  that  much  of  the  ordinary 
conventions  as  to  points  at  infinity 
and  the  ordinary  classification  of 
curves  is  dependent  on  the  choice 
of  the  coordinate  system.  This  fact 
has  already  come  to  light  in  the 
use  of  tctracyclical  coordinates.  The 
present  chapter  emphasizes  the  fact. 

To  obtain  our  system  of  coordi- 
nates take  two  axes  OX  and  OY 
(  Fig.  -1!>)  intersecting  in  O  at  right 
angles,  and  on  each  axis  take  besides  O  another  point  of  refer- 
ence, A  on  OX  and  />'  on  O  Y.  Then,  if  /'  is  any  point  of  the  plane, 
to  obtain  the  coordinates  of  /'  draw  through  /'  a  parallel  to  (>Y 
meeting  OX  in  .17.  and  a  parallel  to  OX  meeting  OY  in  .V.  Let  the 
coordinates  of  M  be  defined  as  in  Jj  *  by 

/-.  •  OM      ./• 

A  = 


/•..  •  AM 
k   .ON 


The  coordinates  of 

writ  ten   as   ( ./•  :  ./•.,,   // 
( 'an  esian  <•< lordinat es 

and    It   recede   to 


/'  mav  then  be  taken  as  (  X.  /u  >  or  otherwise 
//  ).  It  is  cleai'  from  vj  s  ihat  the  ordinary 
are  a  limiting  ease  ot  these  coordinates  as  A 
v. 

l»;  t 


A   SPECIAL  SVSTK.M    OF  ('( )( )KDIN  ATKS  ll',:, 

The  coordinates  bring  thus  defined  for  real  points  the  usual  ex- 
tension is  made  to  imaginarv  points  as  delined  b\  imaginarv  vahirs 
of  the  coordinates.  To  consider  the  locus  at  inlinitv  let  /'  recede 
indefinitely  from  < >.  This  mav  happen  in  three  \\avs: 

1.  J'  may  move  on  a  straight  line  parallel  to  <i.\'.  Then  the  ratio 
./•  :  ./•,  approaches  the  limiting  ratio  /,•  :/,-,,  and  the  ratio  ij  :  i/:  has 
the  constant  value  determined  1>\  anv  point  on  the  straight  line. 

'1.  1'  mav  move  on  a  straight  line  parallel  to  ()  \.  Then  ./•  :  ./;,  has 
the  constant  value  determined  l>y  a  point  on  that  line,  and  //,://., 
approaches  the  limiting  value  /-.. :  /•  . 

•'I.  /'  mav  move  on  a  straight  line  not  parallel  to  U\  or  <>Y. 
Then  .!/  and  X  each  approaches  the  point  at  inlinitv  on  its  respec- 
tive' axis,  and  therefore  the  ratio  ./•  :  ./•,  approaches  /••  :  /.-..  and  the 
ratio  //,://.,  approaches  /•  :  /'(. 

These'  are  the  only  points  which  we  ivcogni/e  as  at  inlinitv.  In 
other  words,  if  /'  recedes  indefinitely  from  <>  it  will  not  he  con- 
sidered as  approaching  a  definite  point  at  inlinitv  unless  the  point 
on  the  curve  approaches  as  a  limit  a  point  on  a  straight  line.  \Ye 
have,  then,  the  proposition 

All  [mint*  <i(  injinttif  h<n-<'  t'nih'ilinutcx  vlt't<-li  xiitixfi/  ///>•  I'l/iKition 

CVi~  Va)^//!-^)^0'  (^ 

To  deline  the  nature  of  the  locus  at   inlinitv  we  note  first  that 

an  eiiiiation  of  the  tvpe 

<V\+  'Vi=     '  ^-) 

if  satistied  hv  real  points,  represents  a  straight   line  jiarallel  \a<>.\; 

and  the  tMiuation 

",//,+  ",.'/,  -'  ()'  (•') 

if  satisfied  by  real  points,  represents  a  line  parallel  to  <>Y.  With 
the  usual  extension  of  theorems  in  analytic  ^eonietrv  we  >av  thai 
these  equations  alwavs  represent  lines  parallel  respectivelv  to  <>.\ 
and  (>}'.  \Ve  must  therefore  sav  that  e(]iiation  (1  )  represents  tvso 
straight  lines  which  have  the  point  (/ji:/,\i,  /••.,:/'()  in  coinnioii.  \\'c 
ha\ 'e.  t  hen,  the  proposil  ion 

Tin'    Ini-llx   iff    illtillit  I/    i-nttx/xfx   nl    tl/'n    xf/'il/i/Jlf    lllh'S    li'li'lll'/  III    i'ii//t  Ill'i/t 

it  IKI'III!  <-iill,'il  ill,'  iluiil,!,'  i><i!itf  (it  intitiiti/. 

The  l'i trefoil i '_;•  discussion  >hows  that  an  important  distinction 
hctWecii  lines  which  are  parallel  cither  to  <t.\'  or  to  OK  and  lines 


Kit) 


T\\  ( >   I  >  I  M  KXSK  )X  AL  G  K<  ).M  KTK  V 


which  aiv  not  so  parallel.  The  straight  lines  which  are  parallel  to 
('A'  or  <  >  }  we  shall  call  ttpeeial  lines  and  divide  them  into  two  fam- 
ilies of  parallel  lines.  Lines  which  are  not  special  we  shall  call 
<>>•>!  ina  r>i  lines.  We  have  already  seen  that  a  special  line  has  a 
point  at  inlinity  which  is  peculiar  to  itself  and  that  all  ordinary 
lines  have  the  same  point  at  inlinity:  namely,  the  double  point 
at  inlinitv.  We  may  accordingly  state  the  following  theorems,  the 
proofs  of  which  are  obvious: 

/.    T'l'o  Kper'nil  line*  <>f  thf  Kii/itf /a  mil  if  1nn-e  tin  point  in  eo/nmon. 

II.  Tti'o  special  linex  uf  diffi't't'iit  fiuiulii'^  in'  a  xjieciitl  line  and  an 
ordinarii  I/HI',  //are  <>nlt/  one  point  in  minnion  icfiie/i  Hex  in  the  finite 
rei/t'on  of  the  plane. 

III.  Tii'<>   nonpamlb'l  ordinari/  line*   h>'tre   alirai/x  the  double  point 
at  infmitif  ami  <>ue  "flier  finite  paint  in  n>/n>n<ni. 

IV.  Tiro   jHiral/el    <>ntin<o\i/    lines    hare    only    tin-    double   point    at 


71.  The  straight  line  and  the  equilateral  hyperbola.    From  the 
pjra=ka'AM, 

I  > 

which  define  the  coordinates,  we  may         .\- 

obtain 

p  (  /-.,./',  —  /-]./'., )  =  kj\.,  '  (>A  —  /r,/".// ; 


E 


• 
i 


Similarly,  OA'  = 


Now  let  6'  (Irig.  •">" )  be  a  fixed  point  with  coordinates 
('ir,:**.,,  /^1:^.,),  l*'t  <'!>  be  the  line  through  C  i)arallel  to  <>Y,  and 
let  <'/:'  be  the  line  through  ('  jiarallel  to  <>X.  Then,  if  the  line  /'.)/ 
meets  ( ' h'  in  M'  and  the  line  /'A' meets  <'l>  in  A"',  we  have 


A   SPECIAL  SYSTK.M    OF  COORDINATES  107 

Consider  now  a  locus  detined  l>v  the  condition 

CM1 

-~  —  const. 
CUV 

rrhis  locus  is  obviously  a  straight  line  through  r,  and  its  equation 

is  of  the  form 

(  a.,./-,       a  r/-, )  (  /-4//j  -  /r;(//., )  -  <M  ^.,//i  -  A^//., )  (  /y1!  -  ^-'V, )  -  ' '.     (  1  > 

where  '/  is  a  constant. 

Conversely,  any  equation  of  the  form  (  1  )  in  which  </  is  not  /cm 

,       «.,  /.',  #,  /• 

or    mtniitv,    and    •-=£--,  ,    represents  an   ordinary   straight 

"i     ^i      &i      k* 
line.    For  ('t,:  a^,  fi_t:{3  )  fixes  a  point  f ',  and  the  equation  is  e<[iii\  a- 


CM'  ...      .  ...  a.,      fc,          (3.,      /r4 

lent  to  -       -  =  const,    it  <(  is  zero,  or  intinitv,  or  "<  or     *=    -, 

c.\"  »•,      /--j          ft      h-A 

the  equation  is  factorable  and  represents  two  special  lines,  one  at 
least  of  which  is  at   intinitv. 

Again,  consider  the  locus  of  /'  detined  by  the  equation 

<  'M'  .  ( 'A"=  const. 

This  locus   is  an   equilateral    hyperbola   with   two   special    lines  as 
asymptotes.     We  shall  call  it  a  */>(•-•/,//  hyperbola.     Its  equation  is 


equation   ('_!)  can  be  factored  and  represents  t  \\  o  special  lines. 

It  is  to  be  noticed  that  equation  (  1  )  is  satisfied  by  the  coordinates 
ot  the  double  point  at  intiiiitv  and  that  equation  (  •_'  >  is  not. 

72.  The  bilinear  equation.  Kquations  (  1  >  and  <  •_' )  of  ^  71  are  of 
the  form 

which  is  a  bilinear  equation  in  ./'  :  ./',,  and  _//,  :  //.,. 

\\'e  shall  no\\  assume  e(|uation  (  1  )  and  examine  11  in  order  to  see 
it  it  is  always  of  one  of  t  he  t\  lies  ol  s  71. 


KiS  TWO-DIM  EXSIOXAL  GEOMETRY 

In  the  first  place  it  is  easy  to  show  that  the  necessary  and  suffi- 
cient condition  that  (1  )  should  factor  into  the  form 

is   that    . I  />  —  /!('=  0.     Furthermore,   the   necessary   and    sufficient 
condition   that   (1)   should   be  satisfied   by   the  coordinates  of  the 

double  point  at  infinity  is 

We  shall  denote  the  left-hand  member  of  this  equation  by  A' and 
make  four  cases  according  to  the  vanishing  or  nonvanishing  of  the 
two  quantities  .A' and  AD  —  />('. 

CASK  I.  AD  —  liC'^  0,  K  -^  0.  The  equation  cannot  be  factored 
ami  the  locus  does  not  pass  through  the  double  point  at  infinity. 
Therefore  it  cannot  be  of  the  type  (1),  §  71.  It  will  be  of  the 
form  (•_!),  §  71,  however,  if  we  can  find  a^  an,  3^  /3.,,  and  a  to  satisfy 
the  equations  a  3  —  ak  k  =  o  I 


These  equations  can  be  solved  by  taking 


a=KC-AI>. 

Hence  ecjuation  (1  )  represents  a  special  hyperbola. 

CASK  II.  AI>  —  IK'  ^  O,  K  =  0.  The  equal  ion  cannot  be  factored 
and  the  locus  passes  through  the  double  point  at  infinity.  We  shall 
compare  the  equation  with  (1),  $71.  The  locus  oi  the  equation 
under  consideration  intersects  OX  in  the  point  (/>:•—/>',  0:1), 
which  we  will  take  as  (a^a,.  /^  :/:?./)•  Tsing  these  values  in  (  1  ), 
^  71,  and  comparing  with  (  1  )  of  this  section,  we  have 


A  SPECIAL  SYSTEM  OF  COORDINATES  100 

whence    a  =  :!'    these    values    agreeing,    since 

—  k.,  k 

K—0.    Since  Al>  —  li<1^  0,  a  cannot  be  /.ero. 

Therefore  the  locus  represents  an  ordinary  straight  line. 

CASK  III.  A/>  —  IK'--  <»,  K  ^  0.  The  equation  is  factorable 
into  the  equations  of  two  special  lines,  one  of  each  family.  Neither 
line  can  be  at  infinity  since  the  locus  does  not  pass  through  the 
double  point  at  infinity. 

CASK  IV.    AD  —  BC—(},  K  —  0.    The  equation  is  factorable  into 

the  equations  of  two  special  lines,  one  of  each  family.  At  least  one 
of  these  lines  must  be  at  infinity  since  tin-  locus  passes  through  the 
double  point  at  infinity. 

I f  we  call  a  singular  bilinear  locus  one  defined  by  the  equation  ( 1 ) 
when  .!/>  —  />('=  0,  and  a  nonsingular  bilinear  locus  one  defined 
by  (1  )  when  .1  />  —  HC  ^  0,  we  have  the  following  result: 

A  nonxini/iiltir  I'itint'itr  lui'iiH  fx  (t  itjjt'cial  hyperbola  <>r  tin  ordinary 
xti'diyht  lint'  according  ox  /(  //<«'x  nnt  or  i(<_>e#  y/</,x.s  throui/h  fin-  double 
pn'int  «t  infinity, 

A  xini/ulur  bilinear  locux  consists  <>f  t/r»  special  /i/icx,  <///»-  <>f  each 
family,  H'ht'i'e  <»n'  <>r  both  <>f  tin'  line*  >nay  bi-  ii  lint1  at  infinity. 

73.  The    bilinear   transformation.     Consider    the    transformation 

7i<,     .,;,'      («A-^^O) 

ffff[=   "sffl  +  ft-jt/Ai 

(«A-ySa7.,^0) 

°"//j^  7-j//i+  o2/A" 

'This  de  lines  a  one-to-one  re  hit  ion  bet  ween  the  points  ( .r^.r,,  //  :  >/ ,} 
and  the  points  ( ./•[:  ./•',  //[:  //', ).  The  following  properties  are  evident  : 

I.  An\r  special  line  is  transformed  into  a  special  line  of  the  same 
family  and  any  singular  bilinear  locus  into  a  singular  bilinear  locus. 

II.  The    lines   at    infinity    may    remain    fixed    or   be    transformed 
into  any   t  wo  special    lines. 

III.  The  point   at    infinity   may  be  fixed   or  be   transformed    into 
any  other  point  either  at   inlinity  or  in  the  finite  part  of  the  plane. 

IV.  If   the   double    point    at    inlinity    is   fixed,   ordinary    straight 
lines    are    transformed     into    ordinary    straight     lines    and     special 
hyperbolas   into  special    hyperbolas. 


170  TWO-DIM KNS10XAL  CEO.MKTKY 

\'.  It'  tin1  double  point  at  infinity  is  transformed  into  a  finite 
point  .1  and  the  finite  point  />'  is  transformed  into  the  double  point 
at  intinitv,  anv  ordinary  line  is  transformed  into  a  special  hyperbola 
through  .Kand  any  special  hyperbola  through  //is  transformed  into 
an  ordinary  straight  line.  The  line  All  is  transformed  into  itself. 

EXERCISES 

1.  Show  thai  the  cross  ratio  of  the  four  [mints  in  which  a  special 
line  meets  four  special  lines  of  the  other  i'amilv  is  unaltered  bv  the 
bilinear  1  rausfonuat  ion. 

"2.  Study  the  transformation  p-''[  =  //,.  p.'1/.  -  -  >/., ,  "•//[  —  ./•,,  fr>/!,  —  .r.2, 
and  also  the  transformation  obtained  as  the  product  of  this  and  the 
bilinear  transformation  of  the  text. 

3.   ( i  iven  in  space  the  hyperboloid  ./•"+  //"  —  ;."-  -  1  and  X  and  /i  de lined 

bv  the  equal  ions 

s  -z        14-  //  •'•  ~  .-:        1  -  .// 

X  =  •      p.  = 

1  —  //       ./•  +  ,-.-  1  4-  //      ./•  4-  ,v 


REFERENCES 

F<ir  the  beiietit  i  >  f  student*  wlin  mav  wish  to  read  nmre  mi  the  subjects 
treated  in  the  t'i  i  rejoin  L;'  text  the  following  references  are  ^iven.  No  attempt 
has  been  made  |n  make  the  list  complete  or  to  include  journal  articles, 
and  preference  has  been  ^iveii  to  books  which  are  easily  accessible. 

I)\i:i:oi  \.   Principe.*  de  irt;onn:t  rie  analyt  ii|tie.    (iiuithier-Villars. 
KI.IIS.   llohere  ( ieoinet  rie.     Lithographed   Lectures.    (iottinuin. 
SU.MOS.  Conic  Sections.     I.nnu;inan->.  (irceii  >v  ( 'o. 
Si  1 1 1  i.  M"deni  Analytical  (icninctry.    '1'lie  .Macinillan  ( 'oinpany . 

I-',  MI  n,  I  nt  rod  net  inn  to  1'rojrciive  ( iconic  try  ami  its  Applications.    John  \Vylic  ^ 

Sons.  Inc. 

M  i  I.M..  ( 'rn>s  Kat  in  ( iciiinctry.    ( 'ambi-iil^e  l'ni\  crsity  1'ivss. 
\"  i  in, i  N  and  YOI  \.,.   I'rojective  ( icninct  ry,  \'ol.   I.    (iinn  and  ( 'oiniiany. 

/'      .    •  re  iitinnnri'iiii-tit  ninl  iinn-Kit'-iiilnni  iimmi-try: 

< '  \  K-  i  \  w.  Nnn-F.iiclidcaii  ( iconic  try  and  'I'rit.r"iionicl  ry .    Longmans.  ( ,  i  ecu  \  ( 'o. 

('ool.ii)!;!;,    Nnll-I-'jlclidean   ( ieonietl'V.     ('larclidon    Press. 
M  \ssis...   NI  in   l-'.nclidean  (i en! net  ry.    <  iinn  and  (  'i  iiiijiaiiy . 

\\',,ii|i».    "Noii   Miiclidi-an    (icoinetry"    (in    Young's    Moiio!_rra]ilis    on    .Modern 
M  al  hciual  ic>).     LnhLTinan.--.  (i  rccn  *v  (  'i  i. 


1'AKT    III.    TIIRKK   DIMENSIONAL    (iKOMHTKV 

CIIAPTKIl  XI 

CIRCLE  COORDINATES 

74.  Elementary  circle  coordinates.  As  the  first  example  of  a 
geometric  element  determined  by  three  coordinates,  thus  leading  to 
a  three-dimensional  geometry,  we  will  take  the  circle.  If  we  con- 
sider a  real  proper  circle  with  the  radius  /•  and  with  its  center  at 
the  point  (  />,  /-)  in  Cartesian  coordinates,  we  might  take  the  three 
quantities  (//,  £,  /•)  as  the  coiirdinates  ot  the  circle.  It  is  more 
general,  however,  to  take  the  ('artesian  equation 

"j  ( .-'•'  +  >/' )  +  ",•''  +  ".,//  +  «t  =  0  (  1  ) 

as  the  definition  nf  the  circle  and  to  take  the  ratios  <i^\  //,:  </, :  »/4  as 
its  coordinates.  The  circle  may  then  be  of  any  of  the  types  specified 
in  ^  .V.i.  If  it  is  a  real  proper  circle  the  coiirdinates  arc  essentially 
the  same  as  (  }/,  /,%  /•'). 

\Ve  may  also  take  the  equation  in  tctracyclical  coordinates  ./•,. 

Vl  +    "..''',>  +   'V:;  +   U  J\=    ^ "  (~} 

and  take  the  ratios  ><  :  u  \  u.,:  u  HS  tht;  coiirdinates  of  the  circle.  If 
the  point  coiirdinates  jrt  are  the  special  coordinates  of  xj  .~>7,  the  circle 
coiirdinates  »(.  obtained  from  ecjitation  ('!)  are  the  same  as  the 
coiirdinates  //,  obtained  from  fijuation  (1  ),  but  in  general  no  sim- 
plification is  introduced  bv  the  use  of  the  special  coiirdinates.  In 
fael,  it  is  in  many  cases  simpler  to  assume  that  the  point  coiirdinates 
./•,  in  {'(illation  (-)  are  orthogonal. 

I'nless  it  is  otherwise  explicitly  statc<l  we  shall  assume  in  the 
following  that  j\  ai'c  orthogonal  tetracvclical  point  coordinates 
connected  by  t  he  rclat  ion  : 


17ii  THI;KK  DIMKNSIONAL  <  ;  K<  >  M  KT  R  v 

As  sho\\  ii  in  jj  ti.'J  the  equation  ot  ;i  special  circle  with  the  center 
r<>  (  y,  .r  )  =  i/^r^  4-  //.,./•.,  +  //;i./8  -f  ///  4       (  '.  (/>  ) 

\\herc,  of  course,  //,  satisfy  the  fundamental  relation  (/>  ). 

Hence,  if  (  '_'  )  is  a  special  circle  the  coetlicients  n  are  exactly  the 
coordinates  of  its  center.  liccause  of  the  importance  of  this  result 
\ve  repeat  it  in  a  t  heorciii  : 

/.  /'.'',  ''/••'  "ft  li'ii/i'tuit  ti't  riii'i/i'lii-ii]  [mint  nx'i/'ih'mifi'K  <iin/  i/t  ///•>•  r//v/V 
,'ni'i'rif/initi'x  biixt'il  n  [mil  fln'//i,  tJn'ii  tin'  circle  rnfit'tlimitcx  ">'  <<  x/n'<-i<il 
•  •//•••I,  '//•<•  (lif  [mint  i-iiui'iltnntt'n  <>f  t/n:  i-fnfi'/'  of  the  r//v/c. 

Two  c'ircles  \\ith  the  eoi'u'dinutes  i\  and  n^  are  orthogonal  when 

?/(/',  tc~)=  >\f<\+  ''./''.,+  'V'':i+  'V'V     "•  O1) 

l-'roin  this  \ve  inav  deduce  the   following  theorems: 

II.  .  1  tint'/tr  fiunt  ion 


I'onril  innfi'K  define*  <<  //nr/ir  <'/'/•<'/>'  cinnple.r  vJiifh  /x  ii<>)Hpnm><l 
"f  ,il!  !•/'/•'•/•  '.s-  ortJiot/onal  t»  <i  fxr>«'  clrdi'  <>  :  <it  :  r?.  :  <i  . 

For  equation  (  7  )  is  simply  equation  (t!)  with  '•,  replaced  by  tin; 
constants  <i.  and  with  u\  replaced  bv  tlfe  \ariablcs  //t. 

The  complex  contains  special  circles  whose  centers  are  the  points 
ot  the  base  circle. 

\\hcn  the  base  circle  is  a  special  circle  the  complex  is  called  a 
xpi'i'inl  complex.  It  consists  of  all  circles  through  the  center  of  the 
base  circle,  and  the  condition  for  it  is 

","4-  "..r-f-  "::+  "42=  0. 

If  '/.  arc  i  he  coi'»rdi  nates  of  the  real  jioint  at  in  fin  it  v,  o<|  nation  (  7  ) 

defines    a    special    complex    consisting    of   all    the    straight    lines    of 
the   plane. 

///.  //  lir.,  ,•//•'•/,  x  !„•/,,>,,/  tn  ,i  l/'nt'iir  <'<>in  j'l'i.r.  nU  ,'!i'<-le»  nf  //,,•  p.'ti'-il 
•  I,  fin,;/  /.//  f/ir  f";,  I,,/,,,,,/  tn  tin'  ,->,„//>  If.r. 

'I  lie    pl'oot    ot    this    theorem    is    left    to    the    student. 
IV.     7'"'"    HI  Hllllttl  Ill-nil*    Iliiiil/'    t'/illilt  tn/ix 

i  /  ii    \-  ii  ii  -f-  <t  ii  4-  ''  "        '  '- 

1      1     '         u     -J    '          ::     .;    '          Ii 

/,  //   4-  fiini+  I  it   4  A  //        "I 

'!>  t'n,    ii    liiniir  rniii/rn,  it,-,,    //•///'•//    I-IIIIK'I  xtx  nt'  it    in  •//'•//  nt    ,-//•'•//.*. 


CIIJCLK   COORDINATES  17o 

To  prove  this,  note  that  the  congruence  consists  <>t  all  circles 
which  belong  to  the  two  complexes  V",",—  0  and  %  /y/i  =  0.  These 
circles  are  also  common  to  all  complexes  of  the  pencil  ot  complexes 


and  is  defined  by  anv  two  complexes  of  this  pencil.  Hut  the  pencil 
(*)  contains  two  special  complexes  given  hv  the  values  of  A.  which 
satisfy  the  equal  ion 

(",  +  X/-,  r  -H  ",+  XA,  )-  +  (  na  +  AA,  r  +  (  >i  (  4-  X/-4  )-  =  0.         (  (.l  ) 

If  the  liases  of  the  two  special  complexes  are  distinct,  the  con- 
gruence consists  of  all  circles  through  two  points  and  is  therefore 
a  pencil  of  circles. 

If  the  liases  of  the  two  special  complexes  coincide,  equation 
(U)  has  equal  roots.  \Ve  mav  without  loss  of  generality  assume 
'V,/;;/|—  (j  to  be  the  special  complex  of  the  pencil.  Then  ^/r  =  '>. 
and  since  ('.'  )  has  equal  roots  'V/f7>i.=  0  :  that  is,  the  point  <it  is  on 
the  circle  ^.  Hence  the  congruence  consists  of  all  circles  which 
pass  through  a  fixed  point  on  a  circle  and  are  orthogonal  to  that 
circle.  Thev  accordingly  form  a  pencil  of  tangent  circles. 

75.  The  quadratic  circle  complex.    The  equation 


defines  a  quadratic  circle  complex. 

Let   >•.  and  "',  be  anv  two  circles.      I  hen  p",=  >'-+  X"1.  is  an\  cir 
of  the  pencil  defined  bv  >•.  and  »\,  and  belongs  to  the  complex   (  1  ) 
when  X  satisfies  the  equation 


/.  Tlii'  Crtiltil  I'lttit'  I'li/lt  il/f'J'  I'n/it  ill  II  X  fl/'n  fJtxfhli'f  "/'  <'»l  lli'l'L  l/t  n  /•«•/(  X 
fl'iillt  illll/  in'iii'll  nt  fl/'i'/i'X  ll/l/i'SX  ill!  f/l'i'/fft  nt  tin'  fnili'll  In  lull/I  (,,  tin 
I'll  III  III  I'.l'. 


174  THKKK    DIMKNS10NAI.   (1KOMKTHY 

Kqiiation  (  •>  )  will  be  satisfied  by  all  values  of  irt  when  r<  satis- 

fies the  equations 

n    >•  +  ii    >•  4-  ''    ''  -+-  "    ''  =  (', 

11    i    '        rj   -2  i;t   3   '       14    4 

/I     >'   -4-  <l     ''    +  "     ''    +  "     ''    =  (K 

a  :1        2I  '  (4) 

</    '•  4-  <t    '•  +  "    >'  4-  <r    r  =  <), 

l;i     1    '         ^:i    i    '         :;:t    ;(    '         114    4 

</    '•  4-  ''    ''4-  ''    ''  4-  <t    >'  =  ", 

14      1     '          'J4     'J  114     :i     '          44     4 

and  any   >\  Avhich  satisfy  these  equations  will  also  satisfy  (1)  and 
hence  he  the  coordinates  ot  a  circle  oi  the  complex.    Therefore 

//.  Am/  '•//•'•//'  ir/tuxi"  miirtliintti'x  r.  tmfixt'//  <<<iii<tfi»nx  (4)  //•///  //*•  « 
i-iri'/i'  <>f'  tin'  <-<>uifih\r  xit<-h  tlutt  <nii/  jtt'iU'il  <>f  r/'/-i'/fx  /r/i/<-//  I'untitinx  i\ 
an<l  <l<n'x  n<>f  lit'  cntirt'hf  <>n  ///*•  fo/tiji/f.r  trill  1ntr<'  »nhf  >\  in  cumnim! 
irith  (/if  <'"/nph'.r. 

Such  a  circle  is  called  a  <?<mf>/<'  circle  of  the  coni[)lcx.  A  double 
circle  does  not  always  exist  in  a  ^'iven  complex,  however,  tor  the 
necessary  and  siii'ticient  condition  that  equations  (4)  should  have 
a  solution  is  that  the  determinant  ot  the  coetticieiits  should  vanish. 
A  complex  that  contains  a  double  circle  is  called  a  xit/</nl<(r  complex. 

If  in  equation)  "2  )  jv  is  the  double  circle  of  a  singular  complex  and 
>/'.  anv  oilier  circle,  of  the  complex,  the,  equation  is  identically  satis- 
tied.  Hence  we  have  the  following  theorem: 

III.  In  n  K/ni/n/iir  coinph'X  tJtr  pencil  <>f  circles  ili'fun'il  />//  tin1  (bntfifi1 
i-iri'Ii'  iiml  mi  ii  nthiT  pcnt'il  "f  tin'  coinpli'x  lit'*  entirely  in  f//f  <'ninj>li',i. 

\Ve  shall  now  proceed  to  find  the  locus  of  the  centers  of  the 
special  circles  of  the  quadratic  complex.  The  special  circles  have 
coordinates  //.  which  satisfy  simultaneously  equation  (1)  and  also 
the  equation  lor  a  special  circle 


The  circle  coJ'irdinates  are  also  (theorem  I,  sj  74)  the  point  coor- 
dinates of  the  centers  of  the  special  circles.  These  coordinates 
define  a  one-dimensional  extent.  Therefore  the  locus  of  the  centers 
of  the  special  circles  of  the  complex  is  a  curve,  which  is  called  a 
<•!/<•/><•  or  a  ///-•//•(•///(//•  fiiri'r  (see  Kx.  i>,  vj  »i^  ). 

Tin-  coordinates  //,  which  satisfy  simultaneously  (  1  )  and  (5)  will 
al>o  sat  isfv   t  he  eipiat  h  m 

^  "  ,  '/,",  +  X  (  //,J  -|-  H-  +  n:  f  //;  )  =  0  (0  ) 


CIKCLK  COOKDIXATKS  17") 

for  all  values  of  X,  and  anv  equation  ot  the  torm  (t!)  in;i\-  replaee 
(1)  in  the  definition  <>f  the  Incus  sought.  But  among  the  com- 
plexes defined  by  ('!)  there  are  in  general  four  singular  complexes 
corresponding  to  the  values  ol  X  defined  by  the  e(|iiaimii 

//  —  X   it       <i       n 

11        r:        1:1        n 

!  ,i        n   -  \   „        n 

«). 
it       a       <i  —  X   <t 

1:1         2:1         :;:;         :;« 

ft       it       it       n  --  \ 

I   14          -4          :il          44 

Hence  we  have  the  following  theorem: 

IV.  Tin'  1'iji-lii'  fa  hi  i/i'tii'fitt  tin-  I'li-nx  at'  tin'  i'1'nfi'rx  nf  tin'  xjii'i'iiil 
fit'i'li'K  nf  n  ii  if  niii-  nt  /<////•  niHi/ulnr  I'linipli'j't'x. 

Take  (  ',  anv  one  of  these  singular  complexes,  and  consider 
the  straight  lines  belonging1  to  the  complex  (  '.  Their  coordinates 
satisfy  a  linear  equation 

('i"i+  'V.J+  '•,.'':;  +  ''4"4=  °« 

where  ct  are  the  cni'irdinates  of  the  real  point  at  intinitv.  Conse- 
quently the  straight  lines  form  a  one-dimensional  extent,  and  by 
theorem  I  anv  pencil  of  straight  lines  contains  two  nf  the  lines  of 
this  extent.  Consequently"  the  lines  of  the  complex  ('  envelop  a 
con  ic.  wliicli  we  shall  call  1\ 

Now  let  />  be  the  double  circle  of  (',  and  '/'  anv  straight  line  of 
<  '  :  that  is,  anv  tangent  line  to  F.  The  pencil  defined  by  f>  and  '/' 
belongs  entirely  to  <  '.  and  consequently  the  tw<»  centers  of  the  two 
point  circles  ot  this  pencil  are  points  ot  the  cyclic.  Furthermore, 
all  points  of  the  cyclic  can  be  obtained  in  this  way,  since  a  point 
ot  the  cyclic  and  the  circle  /'  will  determine  a  pencil  ol  circles 
belonin  to  fund  containin  a  line  /'.  Hence  we  mav  sav  : 


V.  A  ci/i'Itc  i'ii/i  /"•  ilt'tnii'il  (lit/'/  /n  i/i'iirrii/  in  /(////•  /ni  i/a  )  </*  flii' 
/urn*  nf  tin-  rrnfi'rx  nf  tin'  jx'int  i-it't'li'x  "/  tin'  i»')n-ilx  <>f  <'/,;•/<*  ,1,  tin,  •</ 

III/    ll     //./('./    (VV'7,'     1>    ttntl    till'     /il/li/i   lit     /////>'    /"    It     flJ'l'l/    I'nllii'     I'. 

'1  ake  /,'  and  /',,  t\vo  poinls  on  the  conic  I\  and  \vith  /,'  and  /.!  as 
centers  const  1'iict  t\\o  circles  <•  and  ••'  orthogonal  to  /'.  The  circles 
c  and  '•'  determine  a  pencil  ot  circles  orthogonal  to  /'  and  to  the 
chord  /'/.'.  Hence,  by  theorem  \".  ^  til',  if  A  and  .)'  are  the  points 
ol  intersection  ol  <•  and  ••'  ,  A  and  ./'  are  the  centers  ol  the  pom! 
circles  of  the  pencil  of  circles  defined  bv  I>  and  the  chord  /,'/'. 


17li  THKKK    PI.MKNSloNAL   (IKOMKTRY 

Now  let  //  approach  /!  as  a  limit.  The  points  A  and  A'  approach 
.17  and  M'  respectively,  two  points  on  the  envelope  of  the  circles  <-. 
At  the  same  time  .1  and  J'  approach  as  limits  the  centers  of  the 
point  circles  in  the  pencil  of  circles  defined  by  /'  and  the  tangent 
to  the  conic  \\  Hence  we  have  the  following  theorem: 

VI.  A  /•__//< •//••  '•''//  be  'fi'tn'rntt'il  <t*  flu'  <'nreli>]n-  af  a  family  <>f  rird<s 
//•//".»•»•  <••  /tft'/'*  art'  "//  it  ijii'i'ii  1'iinit'  r  'tm/  it'hicJi  art'  nrthotjonnl  f<>  a  t/ivcn 
<-ir<'l>  I >.  J']a<-li  rir<'li'  <>f  tin1  family  is  iJ»ubly  ta>i</<'nt  t»  the  <'t/<-H<: 

This  generation  of  the  cyclic  can  in  general  be  made  in  four 
ways,  since,  as  we  have  seen,  the  cyclic  can  be  obtained  from  the 
point  circles  of  tour  singular  complexes.  The  cyclic  curves  have 
been  exhaustively  studied  both  witli  the  use  of  ('artesian  coordi- 
nates and  with  the  use  of  tetracyclical  coordinates,  but  a  further 
discussion  of  their  properties  would  require  too  much  space  for 
this  book. 

EXERCISES 

1.  <liven    tlu-    ('([nation    2"ifc"i"/t  —  fy   consider   the    polar  equation 
N  'i.f.r-i/,.  =  0.     This   assigns    to  any   circle   a    definite   linear   complex. 
I  >iscuss    this   on    the   analogy   of    polar   lines   with    respect   to  a   curve 
of  second    order  in   the   plane,   defining    tangent    complexes,   self-polar 
systems  of  complexes,  and  the  reduction  of  the  original  equation  to  a 
-tandard  form. 

2.  Prove  that  if  a  quadratic  complex  contains  more  than  one  double 
circle   it   contains   at    least    a   pencil   of  double   circles  and   degenerates 
into  two  linear  complexes  or  a  single  linear  complex  taken  double.     In 
the  former  case  show  that  each  circle  of  the  pencil  common  to  the  two 
complexes  is  a  double  circle  of  the  quadratic  complex. 

3.  If  a   quadratic    complex    degenerates    into  two  linear  complexes, 
>how    that    the   cyclic   defined    bv    it    degenerates    into   two   circles. 

4.  Show  that  anv  circle  in  a  nonsingular  quadratic  complex  belongs 
to  two  pencils  which  lie  entirely  in  the  complex.     Hence  show  that  any 
quadratic  complex  is  made  up  of  two  families  of  pencils  such  that  any 
circle  of  the  complex  belongs  to  one  of  each  of  the  families.     Show  that 
two  pencils  of    the  Name    families  never  have  a   circle   in   common   and 
that  anv  pencil  of   one   family  contains  one  circle  of  each   pencil  of  the 
'  '1  her  family. 

5.  Show  that   the   following  curves  are  special  cases  of  cyclics:    the 
ovals    of    ]>c-cartes    the    ovals    of    ('asshn.   the   cissoid,   the    lemniseate, 

nverse   and   the    pedal   curves   of  conies. 


CIKCLK  rOORDINATKS 


76.  Higher  circle  coordinates.  In  addition  to  the  four  <]iiantiti.-s 
?/ ,  ?/,,  v  ,  a  used  in  the  foregoing  sections,  we  shall  no\v  introduce 
a  tifth  quantity  />.,  defined  \>\  the  relation 

"f  +  "•'+  ";;  4-  "4"4-  >':'=  ()-  (1; 


If  the  point  coordinates  ./•,  used  in  defining  the  elementary  circle 
coordinates  ir  were  not  orthogonal,  \ve  should  deline  i/^  by  the 
equation 


of  which  (1  )  is  a  special  case.  We  may  also,  it  we  ^'ish,  replace 
the  live  quantities  //,  hv  live  independent  linear  combinations  of 
them,  hv  virtue  of  which  equation  (1)  would  be  transformed  into 
a  more  general  quadratic  equation,  so  that  we  may  say  tin-  lii<jhir 
(•//•<•/»•  t'niirdinatex  in  their  i>i»st  t/eneral  _/«/•///  i-iitis/xf  <</'  tin-  r<iti<»<  /</ 
fire  riirin/i/i'x  connected  by  a  fundamental  ijiuadrtitii;  rt'l<tti< 


/i 


We  shall  continue  to  use  the  orthogonal  form  for  simplicity  of 
treatment. 

As  shown  in  ^  •>',)  the  vanishing  oi  the  coordinate  //.  is  the  neces- 
sarv  and  sutVtcient  condition  that  the  circle  should  be  special.  In 
this  case  the  circle  is  completely  determined  bv  the  four  coordi- 
nates n  ,  //.,,  //.,,  M(.  So,  in  general,  the  center  and  the  radius  of  a 
circle  are  fullv  determined  hv  means  of  the  first  four  coi'irdinates, 
a.  ;/.,.  ?/.,,  it  :  that  is,  the  circle  is  completely  determined  in  the 
elementary  sense.  The  absolute  value:  of  //.  is  then  determined,  but 
its  sign  is  not  tixed. 

It  is  neccssarv,  then,  to  distinguish  between  two  circles  which  are 
alike  in  the  elementary  sense  but  differ  in  the  sign  of  the  coordi- 
nate n..  This  mav  be  done  hv  noting  that  anv  nonspecial  circle. 
whether  a  proper  circle  or  a  st  raight  line,  divides  t  he  plane  into  t  \\  o 
portions,  and  hv  considering  a  circle  with  a  fixed  n,  as  the  boundan 
of  one  of  these  portions  and  the  circle  with  a  coordinate  n.  of 
opposite  sign  as  the  boundary  of  the  other  portion.  The  same  result 
mav  be  obtained  bv  considering  the  circle  described  in  opposite 
directions,  with  the  agreement,  perhaps,  that  the  circle  shall  be 
considered  as  bounding  that  portion  of  the  plane  which  lies  on  the 
lett  hand  in  describing  the  circle. 


17S  THKKK    IMMKNSIONAL   CEOMKTKV 

If  ./•   are  the  orthogonal  coordinates  described   in  detail  in  §  T>4, 
that   is.  if  we  introduce  Cartesian  coordinates  so  that 

p.r}  =  J:2  +  >f       1  -      p.':,  =  -  .'-,      pJ\  =  -  //,      P->\  =  —i(  •'•'•'  +  ,'f  +  1  ). 
it    is  easy  to  compute   that    the   radius  of   the  circle   ?/;    is   equal   to 
Hence  to  fix  a  sign  of  ?/.  is  equivalent  to  lixing  the  sign 

",  -'\ 

of  the  radius.  We  may  agree  that  the  sign  ot  the  radius  is  to  be 
considered  positive  when  the  center  ot  the  circle  lies  in  the  area 
hounded  bv  the  circle  and  that  the  sign  of  the  radius  is  to  be 
taken  as  negative  when  the  center  lies  in  the  part  of  the  plane  not 
bounded  bv  the  circle. 

The  angle  between  two  circles  ut  and  r  is  now  defined  without 
ambiguit     bv  the  formula 


or  v  ''  4-  ".,''.,4-  ".,'',,4-  'Y'i^~  ".''.cos  0  =  0.  (2) 

To  change  the  sign  of  it.  but  not  of  rr  is  to  change  the  angle  0 
into  its  supplementary  angle. 

If  the  circles  ",  and  r.  are  real  and  the  coordinates  are  those  of 
sj  i>4,  it  is  not  difficult  to  see  that  the  angle  6  is  the  angle  between 
the  two  normals  drawn  each  into  the  region  of  the  plane  which 
each  circle  bounds. 

If  either  of  the  two  circles  is  special,  0  is  either  infinite  or  in- 
determinant.  In  particular,  if  r.  is  a  special  circle  and  ?/.  is  not, 
we  have  cos  6  —  x  when  the  center  of  rt  does  not  lie  on  ?/.,  and 

cos  0  -  -       when  the  center  of  >'   lies  on   // .     Hence  we  mav  sav  : 
0 

,1   six-i'iill  i-ii'i'li'    tiinh'iK  Ulllj  il>li/lt'  //'/'//I   if  rifi'li'  I'll   t/'/iir/l    ft*   ft'ttfi'f  //Vx. 

Two  circles  are  orthogonal  when  6  ~  -  ( '2  k  4- 1  )  '•  The  necessary 
and  sufficient  condition  for  this  is 

//   r    -\-  H   >•    -I-  ii   /•    4-  //   r          ().  (  '.}  ) 

ll  j    -j  .;    :;  11 

Two  circles  arc  tangent  when  0  (>.  The  necessary  and  suiVicient 
condition  for  this  is 

//  r   -f-  "  ''  4-  "  /'  -\-  »  /•  ~\-  //>'--(}.  I  4  ) 

ll  .'    -J    '        ;;    :;    '        4    i    '        :,    :, 

It  is  to  lie  noted  that  two  circles  are  not  defined  as  tangent  when 
0  -  77.  If  the  circles  are  real  proper  circles  thev  are  tangent  only 


CIKCLK  COORDINATES  17',) 

when  they  ;uv  tangent   in  the  elementary  sense  and  the  interior  of 
one  lies  in  the  interior  oi   the  other. 
Consider  the  equation 

<V'l+  "!'••+   ".;";(  +  ''."4+  "-J'l=   f)  (/">) 

in  the  higher  circle  coordinates.  This  is  equivalent  to  equation  ( '2 ) 
it'  we  place 

it   —  >•  .       a  =  r  ,       n   =  r  ,       <i   =  i'  ,       n   —  r    cos  $, 

1  1 '  ^  -J  ::  a  4.  4  u 

together  with  the  condition 

''?+    ''2+  '':!  +    'Y  +    '5=    °" 

These  equations  are  just  suilicient  to  determine  r.  and  cos  6.  Hence 
t/tf  /I/I//HT  r/rclr  cn///i>lt'.r  i-iinNtntx  nf  circh'x  <'iitttn</  a  fu'>  d  <'ir>'h'  under 
<t  fi.i-fil  itni/li'. 

It'  <t.  -----  II  the  higher  circle  eoinjilex  becomes  the  elementary  com- 
plex consisting  of  circles  orthogonal  to  a  base  circle. 

The  circle  complex  (•>)  is  culled  a  special  complex  when 

"i"'  +  <f-2  +  ltz  +  "4  +  ".-"'  —  (''- 

In  that  ease  #  =  0  and  the  etiuation  may  lie  identilied  with  (4). 
Hence  it  xj>i-i-i<d  ••i>ni[>lt'.r  in  the  lii'/luT  cvunUnittt'S  cnttsixta  ';/'  circle* 
tiini/>  nt  f<i  ii  ti.n'il  <•!>'<•/''. 

Two  simultaneous  eijuatiiuis 

'',",-{-  ''.,^.,+  <'.."., 4-  '',"_,+  <>.».=  0, 
/,  „  +  /,  n  +t,  n  +  I,  u  +ljt   =0 

11'       u    a   '       a    ::    '       -14'       ;,    ;> 

define  a  higher  circle  cntii/ruence.  Circles  \\-hich  satisfy  these  two 
equations  also  satisfy  any  equation  of  the  form 

^(  itf+  X/',. )»/,-=  <l 

liut  amoii'^  the  complexes  deiined  liv  this  last  cijiiiition  are  two 
special  complexes.  Hence  <i  ///////'/•  ••//•<•/(•  cu/ii/ni,  //<>  r^/^.s^.s/.v  (//'</// 

r//V/,  ,v    tilili/Ct/t    tu    (ll'n    t't.K'il    <•!/•'•/>  N. 

EXERCISES 

1.  \Vli;i!     i>    the    confij^linitioil    of   lln-    lii^liel'   circle    eoli^nieliee   it'   the 
t\Vn   >]iecl;d    Ct  >lll]ile\rS    eniuriiie  '.' 

2.  Show  that   if  ,r,  a  i'e  orthogonal   tet  racvelieal  eoiirdiliate^,  t  lie  circle 
eoordiiiates   //  ,  //,,  ?/.,,  n    ai'c  [iroporl  ional   to  the  cosines  of  the  angles 
which  the  eiivle  //,  make-,  \\itli  the  eoiii'dinati'  circles. 

3.  I  >r->c|ilir    (lie   col  11  j  ill  •  X  es   (letillcil    li\    eacll    of   the    equations    n  0. 


CHAPTER   XII 

POINT  AND  PLANE  COORDINATES 

77.  Cartesian  point  coordinates.  Let  < L\\  <>V,  ()Z  (Fig.  ;">!  )  be 
three  axes  of  coordinates,  which  we  take  for  convenience  as  mutu- 
ally ort  ho'_;-onal.  Then,  if  /'  is  any  point  in  space,  and  /'A,  /'J7, 
/'.V  arc  the  perpendiculars  lo  the  three 
planes  determined  by  the  axes,  the 
lengths  of  these  perpendiculars  with  a 
proper  convention  as  to  signs  are  the  M<t 
rectangular  ('artesian  coordinates  of  1'. 
That  is,  we  place 


Ml'.      //=  Ll\      z  =  Xl\     (1  ) 


Fi...  51 


where  .)//',  /. /'.  and  A7'  are  positive  if   <£ 
ineasureil  in  the  directions  O.V,  ()  )",  and 
n'/,  respectively,  and  negative  it  measured  in  the  opposite  directions. 
The  coordinates  may  be  made  homogeneous  bv  placing 


1 


t 


t 


and  taking  the  ratios  x:i/:z:t  as  tlie  coiinliiiatos  of  /'. 

To  anv  point  /'  corresponds  then  a  real  set  of  ratios,  and  to  any 
set  of  real  ratios  in  which  /  is  not  xero  corresponds  a  real  point  I'. 
The  relation  between  point  and  coordinates  is  then  made  one  to 
one  liv  the  following  conventions:  (1)  the  ratios  0:0:11:0  are 
not  allowable:  (  "_' )  complex  values  of  the  ratios  detine  an  imag- 
inarv  point;  ('•'>>  ratios  in  which  /  =  0  but  ./• :  // :  ,r  are  determinate 
detine  a  point  at  inlinitv.  In  fact,  as  t  approaches  xero  /'  recedes 
indefinitely  trom  '  >. 

Il  a  point  is  ii'M  at  infinity  we  mav,  it  \\'e  clioose,  place  f  -=  1 
in  (  '2  ),  thus  reducing  the  homogeneous  coordinates  to  the  in>n- 
hoino^i'iieons  ones.  .Warn,  nonhoinogeiieous  coJinlinates  are  easily 
mad''  homogeneous  liv  di\idin^  liv  /.  Accordingly  we  shall  u>e 

l-u 


POINT   AND   PLANK   COORDINATES  IS] 

the  two  kinds  side  by  side,  passing  from  one  to  the  oilier  as 
convenience  dictates. 

A  more  general  system  ot  ('artesian  coordinates  may  be  delined 
by  dropping  the  assumption  that  the  axes  < L\\  <>  }',  <>/  (  Fig.  ."il  ) 
are  mutually  orthogonal,  and  drawing  the  lines  MI'.  LI',  A7' 
parallel  to  the  axes.  The  coordinates  are  then  called  <,/,/i,jn,'.  Thev 
mav  be  made  homogeneous  bv  the  same  deviec  as  that  used  in  the 
case  of  rectangular  coordinates. 

Throughout  this  book  the  axes  will  be  assumed  as  rectangular 
unless  the  contrary  is  explicitly  stated. 

78.  Distance.  Let  /'  and  I',  be  two  real  points  with  the  coordi- 
nates (./'j,  //j,  ^  )  and  (./-,.  //,,  ,-•„ )  respectively,  and  let  a  rectangular 
parallelepiped  be  constructed  on  /,'/.!  as  a  diagonal,  with  its  edges 
parallel  to  the  coordinate  axes.  Then,  it'  /[/,'.  //>',  and  S/'.  are  three 
consecutive  edges  of  the  parallelepiped,  it  is  evident  that 

/;/,•  =  ./—,•,,     JM  =  yt-ffv     SK=z.2-z,  (1; 

Hence  the  distance  /,'/._!  is  given  by  the  equation 

/;/,;  =  \  '(./•,-  .>•,)-+  < //,—//,  r+  ( z.. - zl r.  (-2) 

or,  written  in  homogeneous  coi'irdinates, 


This  formula  has  been  proved  lor  real  points  onlv.  It  is  now 
taken  as  the  definition  of  the  distance  between  all  points  of  what- 
ever nature.  From  the  definition  we  obtain  at  once  the  following 
propositions : 

7.  Tlif  ilittfiitti'i'  ln'tn-ttii  tn-n  ji'ii/tfx  HI  if/i,  r  •>('  //•/,/',•/,  is  ,/f  iiitlniti/  is 
Jin  it.'. 

II.  Tin  tTtxtitin-f  I,,tir,,  n  it  /mint  lit  intiiiiti/  ,i/i,l  ,/  jL.'int  imt  ,it  intiiiiti/ 
ix  intinitf.  Unli'xx  tin-  //"////  lit  infiniti/  h,is  ,-,  ,,'i/',  I  i  init ,  x  ir!i!<-li  mi/fxt',/ 

_,-'J4-   //'J  -f-  .;•(),  /         O.  (    I   ) 

In    tin'    I'ltttT    nix,'    tin'    ilixtiiiii-,     /:,//'-,,, i    tin-    /,  ..'in!     i/f    intlnit/l    ,Hi'l    'in  >l 
I,, ilnt    nut    lit   ///ti/i/tt/   <x   i  inl  •  t ,  r  in  i  n<tt  <  . 

The  points  whose  coordinates  sati>lv  e([uations  (  \  )  lorm  a  one- 
diineiisional  extent  called  ///.•  .•//•••/-  ,//  ini'miti/.  The  reason  for  the 
use  ot  the  word  '  circle  '  will  appear  later. 


182  THREE   DIMENSIONAL  GEOMETRY 

If  in  equation  {'2)  we  replace  the  coordinates  of  /,'  by  those  of  a 
fixed  point  ('  ( ./•  ,  //  .  ,:  »  and  the  coordinates  of  /'  bv  those  of  a 
variable  point  /'(_./•,  //,  ^  ),  while  keeping  ('/'  equal  to  a  constant  /•, 
we  obtain  /  r  r  ,-j  ,  ,,._,.  \-\(~  _  -.  -\-_-_  t:*  /  .-,  \ 

which  defines  the   locus  of  a  point  at  a   constant    distance  from   a 
fixed  point.     This  locus  is  by  definition   <i  *y///c/v. 
Kquation  (  f> )  mav  be  written   in   the  form 

A  ( ./•-  +  <f  +  r )  +  !>•>(  +  <  'i/t  +  iw  +  i-:t-  =  o,  ( »; ) 

where 

If  the  center  ('  and  the  I'adius  r  are  Unite,  the  coeflicient  A  is  not 
y.ero.  Conversely,  any  equation  of  the  form  (»!)  in  which  A.  is  not 
y.ero  defines  a  sphere,  the  radius  and  the  center  of  which  are  given 
by  (7).  More  generally  it  is  possible  to  define  a  sphere  as  the 
locus  of  any  equation  of  the  form  (li).  In  ease  .1=0  the  center  is 
at  infinity,  the  radius  is  infinite  or  indeterminate,  and  the  equa- 
tion splits  into  the  two  equations  t  =  0  and  />./•  +  ('//+  Dz  +  /,7  =  0. 
These  cases  of  the  sphere  will  be  discussed  in  detail  in  ^  11  S.  In  the 
present  section  we  shall  consider  only  the  case  in  which  A  "--  0  and 
the  sphere  conforms  more  nearly  to  the  elementary  definition,  and 
its  equation  mav  then  be  put  in  the  form  (•">). 

The  I'adius,  however,  mav  be  real,  imaginary,  or  y.ero.  If  the 
i-adius  is  y.ero,  the  equation  takes  the  form 

and   the  sphere  is  called   a  intU  .sy>//<'/v  or  a  /mint  vj>/iff<: 

It  is  obvious  that  if  (./'(i.  //n,  .r  )  is  a  real  point,  equation  (  *  )  is 
>ati>lied  by  the  coordinates  of  no  other  real  point.  There  exist, 
however,  a  doiibK  infinite  set  of  imaginary  points  which  satisfy 
equation  (  s'). 

79.  The  straight  line.  A  straight  line  is  by  definition  the  one- 
dimensional  extent  of  points  whose  coordinates  satisfy  equations 

p.,     :  r, -f  \./-a, 

W      ,'/,+  X//-  (1) 

pz        ?,  +  X-^, 

Pf        ',  +  ^'.,1 


POINT  AM)   PLANE  COORDINATES  IS.'J 

where  (•'', : //! :  ^'[ : ', )  an(l  (•''.,  •'//., : -., :  t ., )  «u-e  tl"'  coordinates  of  t\vo 
fixed  points  and  X  is  a  variable  parameter. 

From  the  definition  we  may  draw  the  following  conclusions: 

I.  AIII/   f ii'n   d/xtinrf   jinintx   <l> •fi'fnn'iif   <l   xtrdii/ht   lull',    nin/   mi//   tiro 
t/txtitn-t  finintx  mi   tin'  ///it-   ni<i]/  in'   HKi'il  t<>  determine   if. 

The  first  part  of  this  theorem  is  obvious.  To  prove  the  second 
part  let  /,'  be  a  point  on  the  line  (  1  )  determined  bv  X  X  and  let 
/.!  be  uuotlier  point  on  the  line  determined  bv  X  =  X,.  Let  a  be 

a   quantity    defined    bv   the    relation    -          — -  —  X.     Then    the    first 

:      .  .  l  +  o- 

e(}iiation  in  (1  )  mav  be  written 

__  j\  4-  X,./-.,  4-  a  ( ./-,  -f  X.,.r, ) 

^1T7~ 

or  r.r  =  ./•;  4-  X,.r.,-(-  a  (.i\  4-  X.,.r, ), 

and  similar  equations  can   be  found   for  //,  z,  and  /.     lint  these  are 

the    equations    of    a   straight    line    defined    by    /,'   and    /;,    \\hieh    is 

thus   shown   to   be    identical    to   that   defined   by    (.i\:  //^  z^.  ( ^    and 

(rs:fy.,:^a:Q. 

II.  A   xtriiti/lit  lini'  cnntalnx  a   «*••//////(•  }>t>/'nt  <tt   infiniti/  unlcxx  it  H<x 
<'tit!r<'l>i  id  ititinitif. 

If,  in  equations  (1 ),  t  —  0  and  /*,  =  (I,  then  f  =  0  for  all  values  of  X. 

Otherwise  /  =  <)  onlv  when  X ---        '»  which  determines  on  the  line 

t., 

the  single  point  at  iniinitv  ^^.,—  -t'J'.>fi_1—yt:zt.,—  zJ:^.  This 
proves  the  theorem.  Straight  lines  which  lie  at  iniinitv  are  some- 
times called  improper  strtwjld  liiu'*:  other  lines  are  called  y»/-<y"  /• 
xt/'iii<iltt  lines. 

III.  If  tll'n  paint*   lit'  <i    xtrnii//it    //in-   tire   rail,   tin1  line   i-nntninx  <!// 
in  fin  if]/  »f  )•<'<>/  i>ii/nfx. 

This  follows  from  the  fact  that  if  the  t  \\  o  real  points  arc  used  to 
determine  the  equations  (  1  ),  anv  real  \alue  of  X  irives  ;i  ]-f;d  point 
on  the  line.  Such  lines  arc  called  /•/<//  ////.'.v.  although  it  should  not 
l»e  forgotten  that  tln'V  contain  an  iutiuitv  ot  imaginary  points  also. 

It  a  real  line  is  also  a  proper  line  we  mav  put  /,.  /,,  and  f  equal 
to  unitv  in  equations  (  1  )  and  \\rite  the  equations  of  the  line  in 

the  form  „       /; 

('2) 


184  TH  li  K K- DIM  KN S I ( ) N  A  L  ( ;  K(  )M  KTK  V 

From  this  and  equations  (  1  ),  £  7S,  it  is  not  difficult  to  show 
tliat  tin1  real  points  of  a  ival  proper  line  form  a  straight  line  in  the 
elementary  sense. 

IV.  An   iniiii/lminf  »tmi>iht  line  may  eontnin  <>ne  real  point  <>r  no 
/•<  ill  i"ii n(. 

To  prove  this  it  is  only  neeessary  to  give  an  example  of  eaeh 
kind.  The  line  defined  hy  the  t\vo  points  (1 :1  :1:1)  and  (1:0:  /:!) 
contains  the  tirst  point  and  no  other  real  point,  while  the  line 
defined  hy  (!:/:/:!)  and  ( 1  :<):/:!)  contains  no  real  point. 
These  statements  may  he  verified  hy  using  the  given  points  in 
(•([nations  (1  )  and  examining  the  values  of  X  necessary  to  give  a 
real  point  on  the  line. 

An  imaginary  line  which  contains  no  real  point  may  he  called 
completely  imitifinary,  one  with  a  single  real  point  incompletely 
iiiHiifiniin/. 

V.  If  f/ie  ilififtim-e  hcticeen  tiro  i>»intx  mi  a  straight  line  /.s  zero,  (Jte 
Jifittinee  between  any  other  ttr<>  points  <>f  the  line  ix  zero. 

To  prove  this  we  may  use  the  coordinates  of  the  points  between 
which  the  distance  is  x.ero  for  the  fixed  points  in  equation  (1). 
Then,  if  1^  and  I',  are  two  points  determined  by  A.  =  X  and  X  =  X, 
respectively,  we  may  compute  the  distance  ![!','  by  formula  (o), 
j  ~S.  There  results 


A  straight  line  with  the  above  property  is  called  a  'minimum  line. 
Such  lines  have  already  been  met  in  the  plane  geometry.  Concern- 
ing the  minimum  lines  in  space  we  have  the  following  theorems: 

VI.  .1  minimum  line  meet*  tJie  plum'  at  infinity  in  the  r//vA-  /if 
infinity,  <in<l,  i-n/irerw/y,  <iny  line  /i"f  /if  infinity  tchich  internectn  the 
<•//•<•/>'  </t  infinity  is  <t  minimum  hue. 

l-'rom  the  proof  of  theorem  II  the  necessary  and  sufficient  con- 
dition that  a  line  meet  the  circle  at  infinity  is 

which  is  also  the  necessary  and  sufficient  condition  that  the  two 
points  (j'-.i/^z^t")  and  (•''.,://.,:,?.,:  O  should  be  at  a  /.cro  distance 
apart.  l>y  theorem  V  the  line  is  then  a  minimum  line. 


1'OINT   AND    I'LANK   COORDINATES  IS-", 

VII.  Throut/h  <tn/j  jim/it  »J  sjxd-c  </<»•><  n  <•"//»•  <>1  iniminiini  ///ex 
u'lui'h  is  <tl*»  <t  fun/if  xji/n'i'f. 

Any  point  in  space  niav  lie  joined  to  the  points  of  tin-  circle  at 
inlinitv.  We  ha\e  then  a  onc-dinicnsional  extent  of  lines  through 
a  common  point,  and  such  lines  form  a  com-  by  definition.  Also 
if  (./•://:  ,r  :/)  is  the  fixed  point  and  (./•://:  z :  f)  is  any  point  on  a 
minimum  line  through  it,  the  coordinates  of  (./•://: .: :  f)  will  satisfy 

the  conation 

(yYo-  -r/T-f  (i/t{i  -///)-  +  (~.t_-  zat)  =  °i  (  :;  ) 

and,  eon\  t'isely,  any  point   whose  coordinates  satisfy  this  equation 
lies  by  theorem  VI  on  a  minimum  line  through  (.r(|:  _//  :  ,r  :  tt  ). 

E(juation  ( :» )  is,  howeyer,  the  eijuation  of  a  point  sphere  in 
homogeneous  form.  Hence  the  minimum  cone  is  identical  with 
the  point  sphere. 

80.  The  plane.  A  plane  is  defined  as  the  two-dimensional  extent 
of  points  whose  coordinates  satisfy  an  equation  of  the  form 

Ax  +  Jty  +  Cz  +  JH  =  Q.  (1) 

From  the,  definition  we  deduce  the  following  propositions: 

7.  If  ttt'o  point x  //''  <>n  it  i>l<tii<',  t/n'  gt)'(t it/lit  l/'/n'  funnt'ctht'/  t/t*//i  //»•.<< 
fntii't'ttj  "/i  tin'  htitt1. 


(./•._, ://.,:  ,;•„:/.,)  satisfy  (1),  then  ( ./^  +  X./-..:  i/{  +  X//..:  .~{ -f  Xr,:  ^  +  \t , ) 
does  also. 

//.  .1  ]>l<tm'  is  uni</Hi'fi/  ilt't>rntiii<<l  l>i(  nnii  f />/•<>'  jmintu  n<>t  "/i  tin- 
x<i/in'  xti'itiijlit  lint'. 

If  ( .r{:  i/^.  .-^  /^,  (,'•,://,:,-,,:/.,),  and  (./;,://.,::.:/.)  are  any  tliree 
points,  the  coefficients  .1.  /.',  <\  and  />  ma\'  he  >o  determined  that 

A.r^/lf/^  C-{+  .Itt^---.  0, 

J./1., +  /•'//,+  C2..+  />'._,=  ()-  (.-  ) 

.(./•.  +  /;//..+  rr.+  /;/a=  0, 

unless  there  exist  relations  of  the  form 

\J\+  \,i\,  +  \->'A=  (l 
X /  +  X.,/.,+  X// ..--.«), 


ISO  THKEE-DIMEXSIOXAL  GEOMETRY 

It  follows  from  throivms  I  and  II  that  any  plane  in  the  elemen- 
tary sense  niav  In-  represented  by  an  (Mpiation  in  the  form  (  1  ). 
The  general  definition  of  a  plane  extends  the  concept  of  the  plane 
in  the  usual  way. 

///.   /'"int.*  (it  intiniti/  lif  in  a  plane  eaUi'il  the  jilane  (it  infinity. 

This  is  a  result  of  the  definition,  since  the  equation  of  points  at 
infinity  is  t  -  <>. 

On  the  plane  ./•  —  0  the  coordinates  //  :  z  :  t  are  homogeneous 
coordinates  of  the  type  of  sj  1  <s.  Similarly,  on  the  plane  //  =  0  we 
ha\e  the  Cartesian  coordinates  j-:z:t  and  on  the  plane  z  =  0  the 
('artesian  eoiirdinates  .r://:t.  (  )n  the  plane  /  =  <>  we  may  deline 
t:y\z  as  trilinear  coordinates  of  the  type  in  ^  'I'l. 

IV.  If  tJiree  pnlntx  iif  a  plane  (ire  real,  tin1  plant'  cvntainx  a  doubly 
in  finite  number  <>f  real  j>»t/i(x. 

From  equations  ('2)  the  yalues  of  J,  />',  <\  and  I)  are  real  if  the 
coordinates  of  the  points  involved  are  real.  Then  in  equations  (\  ) 
real  yalues  may  be  assumed  for  two  of  the  ratios  ./•://:  z  :  t,  and  the 
tliinl  is  determined  as  real. 

Such  a  plane  is  called  a  r>'<il  j>l<ine,  although  it  contains,  of  course, 
an  infinity  of  imaginary  points. 

V.  An//  t/rn  Jlxtini't  }>liitn'x   intersect   in   <i   »trai<jJit   //'/n\   ami  any 
>tt  rail/It  liiu-  null/  In-  Ji'fini'il  ax  the  interned  ion  <>f  ttr<>  i>l<tn<'x. 

('onsider  the  t\\'o  planes 


Thoe  (Mjiiatioiis  are  satisfied  by  an  infinite  number  of  values  of 
the  coordinates.  Let  (  ./'{:  y^.  z^:  t^)  and  (./-..://.,:  ^,:  /,)  be  two  such 
values.  Tlit.-n  the  yalues  (si+\f:y^-}-\yn:zi+\2.i:t-\-\t^  also 
satisfy  the  two  e(juations  so  that  the  two  planes  haye  certainly  a 
line  in  common.  '1  hey  cannot  haye  in  common  any  point  not  on 
this  line  it  the  two  planes  are  distinct,  since  three  points  completely 
determine  a  plane  (  t  heorem  1  1  ). 

A'_;',iin.  a  plane  (  by  theorem  II)  may  be  passed  i  hroU'_di  two  points 
on  a  urivi'n  line  ami  a  third  point  not  on  the  line,  and  two  such 
planes  will  determine  the  line. 


POINT  AND   PLANK  COORDINATES  1ST 

VI.  Any  plane  e.n-<'j>f  f/i>'  filitm1  <i(  infinity  '•'nit<iin*  <i  xini/li'  />/»•  <it 
infinity  i  nn<l  <tny  tti'<>  pin  not  tnfcrx>'''t/><</  in   the  xann'  Inn1  nt  infinity 
a  re  j><ir<il!t  /. 

The  first  part  of  this  theorem  is  a  corollary  of  theorem  V.  The 
second  part  is  a  definitioii  ot  parallel  planes.  'I  he  definition  agrees 
\vitli  the  elementary  detinit ion  since,  by  tlieorem  \',  parallel  planes 
in  this  sense  have  no  finite  point  in  common. 

VII.  An  i/ii<i<ii»uri/  }>/<i»i'  C'lntninx  "/><•  <nnt  »nly  <>n>'  r>i<tl  xtruii/Jit  //>/<. 

Since  an  imaginary  plane  has  one  or  more  of  the  coefficients  in 
its  ('([nation  complex,  \ve  may  write  the  equations  as 

( n i  +  in,,  )./•  4  (  tfl  +  ftf., )//  -f  (  7,  +  iy.,)z  +  (£,  4  /£. ,  )t  -.  -  0. 

This  can  he  satislied  hy  real  values  (./•://:  ^ :/)  \\hen  and  only 
when 


+  7a*  +  o./  =  0; 

that  is,  when  (./•://:.?:/)  lie  on  a  real  straight  line  (theorem  V  ). 
That  the  line  is  real  follows  from  theorem  III,  £  71',  since  the  above 
equations  are  evidently  satisfied  bv  two  real  points. 

The  real  line  on  an  imaginary  plane  may  lie  at  infinity.  In 
that  case  the  plane  is  said  to  be  imnijinnry  <>f  li><iJn>r  "/•'/<•/•.  If  the 
real  line  is  not  at  infinity,  the  plane  is  said  to  be  umti/inary  "f 

VIII.  An  if  jiJiiiu'  nitt'rxt'i-tx  n  xphcrp  in  <t  <'/>•/•/>•. 
Consider  the  intersection  of  the  plane 

and   the  sphere 

n(  .r  4  .//'•  4  r )  4  /'./'  4  '•//  4  «/.:  4  ''/  =  0.  (  1  » 

Any  point  on  the  intersection  of  these  two  surfaces  also  lies  on 
the  intersect  ion  of  (  :\ )  and 

//( .r  +  ,//'4  z-)  4  ( /•  4  X.  /  )s  4  ( '•  4  X />')//  4  ( •/  4  V  ')  - 

\\here  X  is  any  multiplier.     Filiation  (  .»  >  j-ejiresents  a  sphere  with 

the    center 

[(/-  4  X.I):  (--4  X /.'):(•/ 4  Xr  >:  -  -J  ,/ j. 

which  \\ill   lie  in   the  plane    ( -\  )   \\lien 

A.l4'7;4,/r      i^  ,//>  f  (  .1-4  /;:4  < "-  >\  ..    n. 


188  TIIKEE   DIMENSIONAL  GEOMETRY 

The  points  of  the  intersection  of  (•'>)  and  (4)  arc  tliereforc 
shown  to  lie  at  a  constant  distance  from  a  fixed  point  of  the 
plane,  and  hence  the  intersection  satisfies  the  usual  definition  of 
the  circle. 

The  above  discussion  fails  if  the  coefficients  of  the  plane  satisfy 
the  condition  /-_)_  /;-_j_  c-  —  ()_ 

This  happens  for  the  plane  at  infinity  and  for  other  planes  called 
•minium ni  filnni'x.  In  these  two  eases  the  truth  of  theorem  VIII  is 
maintained  l>y  taking  it  as  the  definition  of  a  circle.  This  justifies 
the  expression  "circle  at  infinity,"  which  we  have  already  used, 
and  shows  that  there  is  no  other  circle  at  infinity.  The  ease  of  a 
minimum  plane  needs  further  discussion. 

IX.  An//  jititnt'  n»t  <(  minimum  j>/<t>if  intersect,*  tin-  >•/' /•/•!<•  at  infinity 
in  tiro  jiointx,  U'h'u'li  arc  tin1  <-ir<-lt>  f>ointx  <>f  that  j>fnnt\  A  minimum 
j>l,nn'  ift  tiini/i'nf  (<>  tin-  i-i/'flt'  lit  infinity.  Through  <t//y  j>oitit  in  a  pliine 
U'JiirJi  As  HO(  n  i/itninin/n  /i/iinr  </»  tiro  until  in  n  in  ?i)H'R.  TllTOWjli  <tny 
point  in  (t  minimum  pl<im'  i/ot-x  o/i/y  one  innnnnini  lint'. 

The  plane  (  :> )  intersects  the  plane  at  infinity  in  the  line 
.(./•  -f-  !>>i  +  ('2  =  0,  /  =  0,  and  this  line  intersects  the  circle  at  infinity 
in  two  points  unless  .l~+  f!~-\-  ('"=  0,  when  it  is  tangent  to  that  circle. 
In  the  latter  case  the  plane  is  liv  definition  a  minimum  plane. 

It  is  easy  to  see  that  in  a  plane  which  is  not  a  minimum  plane 
its  intersections  with  the  circle  at  infinity  have  all  the  properties  of 
the  circle  points  discussed  in  £  ^0  and  that  the  metrical  geometry 
on  such  a  plane  is  that  of  ££  45  and  4<>.  The  latter  parts  of  the 
theorem  follow  from  theorem  VI,  ^  "IK 

The  minimum  planes  are  fundamentally  different  from  other 
planes  in  that  a  minimum  plane  contains  only  one  circle  point  at 
infinity.  The  geometry  on  a  minimum  plane  prociits,  therefore, 
many  peculiarities,  some  of  which  will  he  mentioned  in  the  next 
sect  ion. 

81.  Direction  and  angle.  We  define  the  <lirt'i-ti<ni  of  a  straight 
line  as  the  coordinates  of  the  point  in  which  it  meets  the;  plane  at 
intiiiiiy.  This  definition  is  justified  hv  the  facts  that  the  lines 
through  a  point  arc  distinguished  one  from  another  hv  their  direction 
in  accordance  with  theorem  I.  vf  7!',  and  that  a  line  can  !»•  drawn 
through  the  point  with  any  ^iven  direction  hv  the  same  theorem. 


POINT   AM)  PLANK  COORDINATES  IS'l 

We  shall  denote  the  direction  of  ;i  line  by  the  ratios  /:///://. 
Then  \ve  have,  by  theorem  II,  ^  70, 

1:  m:n  =  ./-./,  —  ./•/„ :  //,/j  —  ///„ :  ^/,  —  2/ „, 

where  (./•  : //  :z  :  f  )  and  (.r ,://„:  2,:  /.,)  are  the  coordinates  of  anv 
two  points  of  the  line.    If  neither  of  these  points  is  at   infinity,  \ve 

ma>"writc  /:,,:,=,,^,i:,v^//i:,2-,r 

which    is    in    accordance    with    the    more    elementary   definition    of 
direction. 

From  the  definition  we  have  the  following1  consequences: 

/.   TH'D  noneoin<'idcnt  /hit'*  with  tin'  x/i/m'  diret'tiun  <ir»'  jmrnlh'l. 

Such  lines  lie  in  the  plane  determined  by  their  common  point  at 
infinity  and  two  distinct  points  one  on  each  line  ( theorem  II,  £  *"  ), 
and  they  can  intersect  at  no  point  except  the  common  point  at 
infinity.  Hence  they  are  parallel. 

II.  Tlit1  n>'ft'xx<tnj  <nnl  xi/Jfifii-nf  condition  that  n  line  ohould  I"'  <> 
minimum  line  ix  that  if*  direction  xhould  isntwfy  tin'  condition 

f-+  t>r+  tr=  0. 

This  follows  from  (  X  ),   5;  70. 

In  ij  4*1  we  haye  defined  the  angle  between  two  intersecting  lines 
/  and  /,  by  t  he  e<piat  ion 


where  >»t  and  /»,  are  the  two  minimum  lines  through  the  inter- 
section of  7]  and  /,  and  in  their  plane.  We  shall  continue  to  use 
this  definition. 

Now.  if  the  lines  /,.  /  .  HI     and  /;/ ,  intersect  the  plane  at  infinity  in 
the  points  A  ,  /,,,  .17    and  .17,  respect  ivelv,  we  have,  by  theorem  I.  vj  1  ti. 


From  this  we  haye  the   following  theorem,   in   which  the  condition 
that   /!  and   /,  should   be   intersecting  lines  may   be  dropped: 

III.     Till'    'lU'lli'    ln'tll't'1'll     t'l'"     lini'X     ix    ,</>/<!/    fu     f//,'    }l)''iji'ff!l'i'     d/'xfllHi'i' 

ln'tn'i'i'n    tin'   i>i>uitx    in    //'///'•//    tin  i/    n/ti'/'xi  •  t    tin'    ii/iiii,'    <it    iiit:nilii.   tin- 
i-if-fi'  'if  intiittt//  /"  lii'i  tnl:,  a  ,fx  tli,    fu  ii,lii  in<  ntiil  --.////<•  «//<//  tit,-  C"t/xt'ttif  /\ 


!<.)()  THKKK    1MMKNSIONAL   CKO.MKTKY 

The  cross  ratio  (  I.J..,,  -l/r'/,)  is  unity  when  and  only  when  .1^ 
and  .V,  coincide  or  /.  and  A  .coincide,  it  being  assumed  that  neither 
/.  nor  /.,  lies  on  the  circle  at  intinity.  In  the  former  case  the  lines 
/  and  /,  are  parallel  :  in  the  latter  case  they  lie  in  the  same  minimum 
plane.  Hence  follows  the  theorem: 

IV.  It' t  n'n  nmi  mi  ni  mn  ni  Inn'x  ure  mtrnUi'l  <>/•  if  they  Jn'  in  the  sm/ie 
minimum  i>/<ini-,  t/n'//  innke  <t  ,r<r"  tin>//>'  tcitli  each  other,  nn<l,  cnn- 
I't'rxt'fi/,  if  hi':'  ni'itininimnm  ////<•*  nui/ce  <t  zi'/'o  mii/li-  tt'/f/i  each  other, 
t/ii'i/  '!>''•  i/'t/n'r  [niriillil  m'  III'  in  tin'  xiime  minimum  ji/tiin. 

Let  us  suppose  that  /  and  /.are  nonminimum  and  distinct  and 
that  their  directions  are  A  :  I! ^  (\i\m\  .(.,:  />'.,:  < '.,  respectively.  Then, 
as  in  (J  ),  $  41\ 


From  this  we  obtain  the  following  result  : 

V.   Til'"    nnnminimnm    Hn<'x   <tr<'   perpendicular   /<>   <'<(/'/i    other 
their   dire<'ti»ns   itiitiaif   tin'    <-"inliti<'n 


Interpreted  on  the  plane  at  infinity  this  means  that  the  two 
points  (-•!.:  /',:  <  \  )  iind  (  ./.,:  //,:  < ', )  lie  each  on  the  polar  of  the  other. 

VI.  If  .l./'4-  /'// 4-  ('?  +  I >t  ----.()  is  n<'t  n  minimum  pl<i>n',  mil/  line 
irith  tin1  il n'i''-t i'in  .1:  /.' :  ('  <l»,x  ii"t  Hi'  in  tin'  ji/inie  nml  ix  pi'l'i'i'il- 
<li<-iil<tr  t"  i'1'i'i'il  1 1 in-  in  tin'  iiliim. 


The  plane  mentioned  meets  the  plane  at  infinity  in  the  line 
A.r  4-  /•'//  4-  ('-~  --=  °.  :uid  any  line  with  the  direction  .  I  :  /.' :  ( '  meets 
t  he  plane  at  in  tin  it  y  in  the  point  (  .  I  :  /.' :  (' ),  which  is  t  he  pole  of  t  he 
line  .(./•  4-  /.'//  4-  ( '-;  -  0  with  respect  to  the  circle  at  intinity.  Hence 
the  point  (  .1  :  /•':  <'  )  will  not  lie  in  the  line  .  /./•  4-  l>;i  +  <":  -  -  <>  unless  t  he 
latter  is  tangent  to  the  circle  at  infinity.  This  jn-oves  the  theorem. 

Any  line  with  the  direction  .1:  /.' :  ('  is  said  to  be  u<>rnnil  to  the 
plane  .  /./•  4-  />'//4-  ' '.?  4-  I >t  ---  ".  and  tliis  <lesignation  is  used  smnet  imes 
even  tor  minimum  planes.  The  above  discussion,  however,  estab- 
lishes t  he  foHmvinLT  t  heorem  : 


POINT   AND    1'LANK   <  '(  X  >K  I  >I  N  ATKS  I'.l 

By  (  1  )  a  line  with  the  direction  /  :  m  :  n  makes  with  the  axes  o 
coordinates   the   angles   n.   /:?,    7,    \\  here 


,  in 

cos/:}"  '     cos  7  = 

\  l~+  iii--\-  it'2  \ 


These  (|iiant  it  ies  are  called   the  tlirt'<'ti»n   i'nxiiH'1*  <>t    the   line. 
\Vilh  their  use  equations  (jl)  of  ^  7'.'  may  he  put  in  the  form 


where  it  is  easy,  to  show  that  r  is  the  distance  of  the  yariahle  point 
(  ./•,  //,  .?)  from  the  fixed  point  (.r^  //r  ^1  ).  It  is  oh\  ions  that  these 
equations,  do  not  hold  for  a  minimum  line. 

EXERCISES 

1.  Show  that    through    any   imaginary  point    in    space    there   imes   a 
]M.Mieil  of  real  planes  having  a  real  line  as  axis. 

2.  Show  that    the  equation   of  any  imaginary   plane  of  lower  order 
may  he  written  n.r  -f-  /'//  +  ''.'-'  +  </f  —  '*.  where  n,  />.  and  <•  ai'e  real  and  '/ 
is  complex. 

3.  Show   that   any   imainnary    straight    line   either    lies    in    one    real 
plane  and  contains  one  real   point,  or  lies  in  no  real  plane  and  contains 
no   real    point.     The   last    kind    of   lines    is   called   rump!,  /<•///   i  ut'/ifi  n<tr// 
and   the    former   kind    !nfi>iii^/t'ff/t/   i  mm/i  nn  n/. 

4.  Show  that,  the  necessary  and  sntlicient   condition  that  two  points 
should  determine  an    incompletely   imaginary   straight    line   is   that    the 
two  points  lie  in  the  same  plane  with  their  conjugate  imaginary  points, 
hut   not   on  the  same  st  raight   line. 

5.  Show    that     two    conjugate    imaginary    points    determine    a    real 
straight   line  and  that   it   an  imaginary   point    lies  on  a  real  straight    lim- 
its conjugate  imaginary  point  does  a  No. 

6.  Show  that    a    minimum    line    makes    an    infinite    angle    with    any 
other  line  not   in  the  same  minimum    plane   with  it   and   make-,  an   inde- 
terminate angle  with  any  line  in  the  same  minimum   plane  \\ith  it. 

7.  If  (  '_'  i  is  taken  as  the  definition  of  perpendicular  lines,  show  that 
a  minimum  line  is  perpendicular  to  itsell  and  that  a  line  in  a  minimum 
plane  is  perpendicular  to  cyer\    minimum  line  in  the  plane. 


1 ! ivj  Tin;  K i :  i  > i  M  KN s i <  >N  A  L  c;  K< >M  I«:TR v 

8.  If   tin-    angle    between    two   ]ilancs    is    llit'    angle    between   their 
normals,  show  that    two  iioimiinimuni   planes   make  a  xero  angle  when 
thrv  ar>'  parallel  or  intersect    in  a   minimum  line. 

9.  Show  that  anv  minimniii  plane  makes  an  infinite  angle  with  any 
plane  not   intersecting  it  in  a  minimum  line  and  makes  an  indeterminate 
angle  with  any  plane  intersecting  it  in  a  minimum  line. 

10.  Show   that    the   coordinates   of  a   point,  on    the   circle   at    infinity 
can   lie  written  ./•  :  //  :  :.  —  1  —  .s-'J :  /(  1  -f-  N'J)  :  '1  N,   where  ,s  is  an  arbitrary 
parameter.    Hence  show  that  the  equations  of  a  minimum  line  may  be 

written 

./•  =  ./-j  -f  (  1  —  .s- )  /-, 

//  =  //!+''(!  -f-  *-)  /•, 

z  =  zl  +  '2«r, 

where  x  is  fixed   for  the  line  and  /•  is  variable. 

11.  Show  that  the  equations 


where  F('N')  is  an  arbitrary  function,  represent  a  minimum  curve;  that 
is,  a  curve  such  that  the  length  between  any  two  points  is  zero  and 
the  tangent  line  at  anv  point  is  a  minimum  line. 

12.  Show   that    a    minimum    plane    through    the    center    of  a    sphere 
intersects  the   latter  in  two  minimum  lines  intersecting  at  infinity. 

13.  If  a  line  is  defined  bv  the  two  equations 

V+  '\H+  ('r+  'Y  =  °> 
-I.,'-  +  /•'.,//  4-  <".,,'-•  +  1>J      <>. 

show  that  its  direction  is  /,y .,—  ]],f\ :  <\.  I .,—  r .,.  I , : .  1 ,/.'.,—  .-1 J^. 

14.  Sliow  by   reference  to  the  ]ilane  at    infinity  that    the  necessary 
and  -iiflicient  condition  that   the  plane  .!./•  4-  /;//  -f-  ('r.  -f-  I >f  =  0  should 
be  parallel  to  a  line  with  direct  ion  / :  ni  :  n  is  .!/-(-  /-'///  4-  ' ' »  =  0. 

15.  Show  t  hat  the  equal  ion  of  a  plane  t  h  rough  the  point   (,/•  :  //  :  ::  :  f  ) 
and   parallel  to  the  two  lines   with  the  directions  /  :  //>   ;  n    and  /,:  ///,:  //.,, 
respect  ivelv,   is 


POINT  AND   PLANE  COORDINATES 


11  »3 


82.  Quadriplanar  point  coordinates.  Let  us  assume  four  planes  of 
reference  AI'><\  AT>I>,  AI><\  and  lU'D  (  Fig.  ~>-  ),  not  intersecting  in 
a  point,  and  four  arbitrary  constants  /• ,  /,-,,  /•,,  h^.  Let  j>  ,  y-,.  y  .  y/ 
be  the  lengths  of  the  perpendiculars  from  anv  point  /'  to  the  four 
planes  in  the  order  named,  the  sign  of  each  perpendicular  being 
positive  or  negative  according  as  /'  lies  on  one  or  the  other  (arbi- 
trarily chosen')  side  of  the  corresponding  plane.  Then  the  ratios 

.>•    :  f  :  .r  :  .r  =  /,-  /<   :  /•  />   :  k  />   :  /,'  /• 

l         -j         :;        4  l/i         .'  j         ,;I  .',        4/4 

are  the  coordinates  of  the  point  /'. 

It  is  evident  that  if  /•"  is  given  as  a  real  point  its  coi'irdinates  are 
uniquely  determined.  Conversely,  let  a  set  of  real  ratios  ./  :  ./•  :  j  : ./ 
be  given,  no  one  of  which  is  /.ero.  The 
rat  io  ./•  •  ./'4  is  one  of  the  coi'irdinates  of 
any  point  in  a  definite  plane  through 
IS<\  and  the  ratio  -/;,:.?•  is  one  of  the 
coordinates  of  any  point  on  a  definite 
plane  through  /!/>.  The  two  ratios  are 
part  of  the  coordinates  of  anv  point  on  a 
definite  line  through  /.'  and  of  no  point 
not  on  this  line.  Call  this  line  /.  The 
ratio  ,/•,:./•  is  one  ot  the  coordinates  ot 
anv  point  on  a  definite  plane  through 
1  '[>.  ('all  this  plane  in.  If  the  plane  m  and  the  line  /  meet  in  a 
point  /',  the  ratios  ./•  :./•_,:./•.: r  have  fixed  a  definite  point.  If  the 
line  /  and  the  plane  ///  do  not  intersect,  we  shall  say  that  the  ratio> 
define  a  point  at  infinity. 

Complex  values  of  the  ratios  define  imaginary  point--,  and  the 
ratios  0:0:  0  :  0  are  excluded. 

It  one  ot  the  coordinates  is  /.ero,  the  other  three  are  trilincar 
coi'irdinates  on  one  ot  the  planes  of  reference.  For  example,  it  ./•.  0 
the  ratios  j- 1:  j\  :  ./•  arc  tri linear  coordinates  in  the  plane  .  I  /'-<  .  since 
the  distance  of  a  point  in  the  plane  .\l><  from  the  line  .('  is  equal 
to  its  distance  from  the  plane  A<'1>  multiplied  by  the  cosecant  of 
the  angle  bet  ween  the  planes  .  J  />'<"  and  .!/.'/>.  and.  similarly,  for  t  he 
distances  from  A  /'  and  /.'' '. 

1 1  ence  all  values  of  t  he  rat  i os  ./.:./:./  :  ./•  .  except  t  he  unallow- 
able, ratios  0:0:0;  ".  determine  a  unique  point. 


104  T1IKKK   DIMENSIONAL  GKOMKTRY 

Referring  to  the  figure,  we  note  that  -''j=  "  <>n  the  plane  .(/>'"; 
./•..  (I  on  the  plane  .!/•'/>:  ./'.,=  0  on  the  plane  A/H'i  and  j^=  0 
on  the  plane  />/!< '. 

The  point  .1  has  the  eni'inlinates  0:0;0:1,  the  point  It  the 
coordinates  0  :  <>  :  1  :  <>,  the  point  < '  the  coordinates  0:1:0:  0,  the 
point  f>  the  coordinates  1:0:0:0.  The  ratios  k^:k.,:k.^k^  are 
determined  by  the  position  of  the  point  /,  for  which  the  coordinates 
arc  1:1:1:1,  and  this  point  can  be  taken  at  pleasure. 

Qnadriplanar  coordinates  include  ('artesian  coordinates  as  a  spe- 
cial or  limiting  case  in  which  the  plane  ./•  =  0  is  taken  as  the  plane 
at  infinity.  For  if  the  plane  !'><'!>  recedes  indefinitely  from  A,  and 
the  point  /'  is  not  in  !'><'!>,  the  perpendicular  />t  becomes  infinite  in 
length,  but  /,"4  can  be  made  to  approach  zero  at  the  same  time  and 
in  such  a  manner  that  lim /.*4/'4  =  1.  Finally,  if  the  planes  AI>(\ 
.!///>,  and  A<'I>  are  mutually  orthogonal  and  /"t  =  7r.,=  /r.t  =  1,  the 
coi'irdinates  are  rectangular  Cartesian  coordinates. 

If  the  planes  J//<".  A/!/>.  and  A<'I>  are  not  mutually  orthogonal, 
we  mav  place  fc}  —  csc'^,  where  n  is  the  angle  between  A  /!  and  the 
plane  A<'I>,  and  take  similar  values  for  /.-..  and  /-...  We  then  have 
oblicpie  Cartesian  coi'irdinates. 

In  nsin<4  cniadriplanar  coi'trdinates  it  is  not  convenient  or  neces- 
sary to  specify  the  coi'irdinates  of  a  point  at  infinity.  In  fact,  such 
points  are  not  to  be  considered  as  essentially  different  from  other 
points.  Distance  and  all  metrical  properties  of  figures  are  not 
conveniently  expressed  in  terms  of  qnadriplanar  coordinates  and 
should  be  handled  by  Cartesian  coi'irdinates.  \Ve  may.  however, 
pass  from  the  general  qnadriplanar  coi'irdinates  to  ('artesian  coi'irdi- 
nates by  simply  interpreting  one  of  the  coordinate  planes  as  the 
plane  at  infinity. 

83.  Straight  line  and  plane.  We  shall  prove  the  folio  wing  theorems  : 

I.  If  //  :  //„  :  i/ , :  //  <nnl  z  :  z /.  z  :  z  '//•<-  /'/•»  ti.ri'J  jxihitx,  ///>  ,;,,','/•<//- 
itittfit  <>f  nnif  pn'uit  <>)i  fin  xfr<ii'/ht  ////»•  j'oinitn/  tln'in  ///v 


TOINT   AND    1M.ANK   COORDINATES  T.lT, 

This  is  the  definition  of  a  straight  line  for  imaginary  points.  If. 
however,  the  points  //(  and  ;t  are  real,  the  points  given  1>\  real 
values  ot  A.  are  real  points  which  lie  on  a  real  straight  line  in  the 
elcinentarv  sense.  This  is  casilv  verified  hv  the  student  in  using 
a  construction  and  argument  similar  to  that  used  in  ^  '23  for  the 
straight  line  in  the  plane. 


This  is  the  definition  of  a  plane.  If  //,  and  .~:t  arc  anv  two  points 
satisfving  the  equation  of  a  plane,  the  coi')rdinates  of  anv  point  on 
the  line  joining  //-  and  ,?,  also  satisfv  the  equation  :  that  is,  the  line 
which  joins  anv  two  points  of  a  plane  lies  entirelv  in  the  plane. 
Hence,  if  the  plane  contains  real  points  it  coincides  with  a  plane 
in  the  clciiiciitarv  sense. 


///.   Three  points  n<>t  in  the  x<n>t<'  strdujht  l/iif  dcfcfi/nnf 
mil  if  one  plant'. 

The   proof   is   as   in    £  ,S().     If  (//(,   zt.   ti  are    the    three   points,   th 
equation  of  the  plane  is 


(        t         t         t 

i  -j  .;  i 

IV.    //'   i/t,    Zr    ilinl    fi    lire    Ktl//   three   jmhittf   >u>t    «n    tin'    xitnie    xt)'itii/ht 
///H',    the    <-<i(i/\(iti(<t('X    <>f   'Hi  l/   [>o'tnt    oil    the    j>l<tne    t/l/'uU<//i    t  In  lit    imllj    ne 

written 

p.r  -  -  ii  +  X.r  +  ut  . 
r    \       ,'  i    '         i   '    ~  i 

p.i  ,    -  //   -\-  X^.,+  P-t  ,• 


I'.Hi  THKEE   DIMENSIONAL  (JEOMKTHV 

V.  Ami  ('/'a  tlixtinct  iilttiii'x  i  itte/'xcrt  in  a  gtml</ht  line. 

The    proof   is  the  same  as  that   of  theorem  Y,  £  SO.     A   line  can 
therefore  lie  dctined   l>v  two  simultaneous  equations  of  the  form 

ITT       7  A-      X^  7       'V  ^  7 

VI.  11        /,*',•<'  '      (tll('        7    ''; 

^A 

is.  t'"/'  ((n  if  fit! ni'  at'  X.  tftc  <'<fi«ttn>n  iif  (t  phine  thrmi</li  the  line  of  ui- 
t,  rn<  i-timi  "f  tin'  ///>•/  tiC'i  /il<(nt'N.  Ax  \  td/ct'N  it//  t'd/ uex,  all  j>ldncx  of 
tin'  [n  in'il  null/  l'i'  "lit <ti /tt'tl. 

VII.  .I////  f/t/'i'i'  i>l<tni'x  nut  bi'lmii/i ni/  In  (//<•  Mi/tie  pencil  tnfi'/'xt't't  in 

To  prove  this  consider  the  tliree  ('((nations 

V1+V,+  V;!+V,-<>' 

<\-'\+'Y,+  'Y,+  'Y^(}- 
These  have  the  unique  solution 


•2  a  4 

./•:./•:,•:,•=:    I,      I,.     I, 


'\  '4        'l         '-j      '  'i        '»        ';i 

4         1  4         i         -j  I        2         a  i 

unless   the  determinants    involved   are   all   y.ero.     I>ut   in   the   latter 
case  there  must  exist  multipliers  X,  p.,  p  such  that 

and  hence  the  tliree  planes  belong  to  the  same  pencil  l>v  theorem  \  I. 

/,/,1111'S     lint     ll'tlinijilKj    I"     till'    Xllllll'    pl'llt'il,     t/lt'/l 


/.v  tin    fifintfi'i/i  "f  a  [ilitin"  tli  fniii/li   flu'ir  /mint  if  tut,  /-.•••<  «7/"//. 

<///-/  /^   /-//,•(•  (///  rnhirx.  'ill  filitnrx  ////''///'///   '?   i-"iitiH"it  jinint   i-ii  n  !><•  J 

>'//••//    lillllti'X    f'n/'/tl    <l    /'/'//'//'•. 

The  pi'i  K  if  is  i  il  ivi<  His. 


.  lx  X 


POINT   AM)    1'I.ANK  COORDINATES  T.I7 

84.  Plane  coordinates.  Tin-  ratios  of  the  coefficients  in  the  equa- 
tion  of  the  plane  art-  sufficient  to  fix  the  plane  and  inav  lie  taken 
as  the  rtit'irili/Hifi'x  <>f  fin-  j>l<rti>'.  \\  e  shall  denote  them  1>\  i/t  and  sav 
that  it  :  /'.,:  n.^:  i<4  are  the  plane  coordinates  of  the  plane  whose  point 

equation  is 

Vi+  'Vs+  ?V'a  +  W        '  (^  ) 

No  dit't'ereiiee  is  made  in  this  delinitioii  if  the  point  coordinates 
are  Cartesian.  Equation  (  1  )  is  the  condition  that  the  plane  ui  and 
the  point  xi  should  lie  in  tniitnl  //«.v/V  /--//;  that  is.  that  the  plane 
should  pass  through  the  point  or  that  the  point  should  lie  on  tin- 
plane. 

\Ve  have  the  following  theorems,  whieh  are  readily  proved  l»v 
means  ot  tin  >se  ol  vj  So  : 


pi/   —  r  -(-  \/t'  . 


^/i'7  (///v  i>lnn>'  iritli  tJicxi'  I'oflrilinutex  fKtxsi'x  tJtt'nui/Ji  iJnx  linf. 

The  proot  is  obvious.  Equations  (~2)  ai'e  the  equations  of  a 
peneil  of  planes.  They  are  also  called  the  j>l<in<:  <V/;M//«//X  <>f  <i 
^trail/lit  line,  the  axis  of  the  pencil.  In  this  method  of  speaking 
the  straight  line  is  thought  of  as  earrving  the  planes  of  the 
pencil  in  the  same  sense  as  that  in,  which  liv  the  use  of  equa- 
tions (1),  £  So,  the  straight  line  is  thought  of  as  carrying  the 
oints  of  a  rane. 


/x  xittixtli'tl  l>i(  tin'  funfilinntfn  "f  all  floiii-K  tln-'>ii,  ih  ,/  //'./,,/  ///.////. 

It  follows  from  (1  )  that  all  planes  \\hose  eoi\r<liuales  satisfy  ('•'•) 
are  united  \\'  it  1  1  the  point  it  :  it  /.  *i  :  <i  .  ICijUat  ion  (  •'•  )  i^-  t  h<'i'etore 
called  the  plane  equation  of  the  point  it  :  //,:  <r.,:  <t  ,  in  the  same 
sense  in  which  equation  (  -  ).  ^  s:'>.  i>  the  point  equation  of  tin- 
plane  <t  :  (i  :  it.  :  <<  . 


108  TIIKKIv  DIMENSIONAL  GK<  >.M  KTK  V 

III.   Thi-t-e  planes  ii'it  helon'iiii'i  t"  the  same  pencil  determine  <i  point. 

'I'his  is.  nf  course,  the  sanu1  theorem  as  \"II,  ^  S>5,  hut  in  plane 
coordinates  we  prove  it  l>y  noticing  that  tlnve  values  of  ?/,.,  say  r, 
irt,  *,,  which  satisfy  (^"> )  arc  sul'ticient  to  tletennine  the  eoeflieients 
of  ('•'>)  unless  psi-=\ri  +  ^/r;.  The  equation  of  the  point  determined 
1>\  the  three  planes  is.  then. 


4   -0.  (4) 

//•       ^*       //•       //' 


IV.  If  ?',.,  ?/',.,  and  ni  lift-  an  i/  three  planes  )i/>t  l>el»n</ln<j  to  tin-  same 
pi'itt-i/,  the  coordinates  <>f  <itt>/  p/iitte  fhrvw/h  their  cu//t//i"/i  paint  ttre 

put~  i\ . -f-  \//v -(-  ^.sv, 

*///'/  (/////  j>f(ine  U'tth  tJtexe  coordinates  p< taxes  tJirow/Ji  t/iix  jmint. 
The  proof  is  olivious.    These  jilanes  form  a  f>n/t<l/e. 

V.  Tiro  linettr  equations  ic/tifh  are  distinct  << re  satisfied,  lij  the  enurdi- 
nafes  i  f  j>l<ines  u'JiicJi  //ass  tlirouyli  a  atnwjlit  litu'. 

This  follows  from  the  fact  that  each  eipiation  is  satisfied  l>v 
planes  which  pass  through  a  fixed  point.  Simultaneously,  therefore, 
the  equations  arc  satisfied  by  planes  which  have  two  points  in  com- 
mon, and  these  points  are  distinct  if  the  equations  are  distinct.  The 
planes,  therefore,  have  in  common  the  line  connecting  the  two  points. 

The  equation  of  a  straight  line  can  therefore  he  written  in 
plane  coordinates  as  the  two  simultaneous  equations 

</  //  +  ".,"., -f-  '',.".  +  a  n  =  ", 

V'l  +  V's  + 'Va  + 'V<4  =  °- 

VI.  If    "V'/,",—  ')    and    ^>  /^.H.—  (.)    are    tJie  plane    i'<f>iat!>>/ts    of  two 
points  not  <-<iiti<-id<'iit<  then  V^'///,-f  X'V/y/^--  I)  i*  ill,'  plane  equation  ,,f 
an//   1'otnt   on    tlif    Inn'   i-o/inerttni/  fin-   (irsf   (//'»  fio'mts.     As  X  t<ik<  x  all 
>'ii///i'x,  all  points  ol   a  ra/ii/e  can  //»•  (Jinx  ol'tatned. 

VII.  If  ^'t,'!,--  '»,  V/v/.=  0,  and  ^<;",       0  are  the  plane  ,  ^nations 
of  t  ft  ree  points  not  in  //,<•  same  plant',  f //•'//  ]V",-",-+^-?  n,lli'^~fj-^  '',",    -  " 
is  tJie  plane  etptatinn  "t'  ami   //"////  «n  fj/e  plain1  diti'nuiind  I, if  tin1  first 
thr<>    points.     As  X   <///</  fj.   tah','   all   i-af/ii's,   all  jioints  ,<//    //,,•  plane  /-an 


POINT   AND    PLANK  COORDINATES  l'tl»l 

The  proofs  of  the  last  t  \\  o  theorems  follow  closely  from  theorems 
I  and  II  of  :<  s:>. 

'1  he  theorems  ol  this  section  are  plainly  dnalistic  to  the  theorems 
of  the  previous  section.  \Ve  exhibit  in  parallel  columns  tin.'  funda- 
mental (lualistic  objects  : 


I'n  hit 

Points  in  a  plane. 
l'i lints  in  1  \vo  planes. 
A  strai^lit  line. 
Points  of  a  raiiu'e. 
Planes  of  a  bundle. 


riant- 

Planes  through  a  point. 
Planes  through  two  points. 
A  st  rai^'ht  line. 
Planes  of  a  pencil. 
P<  'ints  of  a  plane. 


EXERCISES 


r  ./•   4-  /•  ./•    +  /•  ./•   —  0, 

-2     -J      1^       :!     :i      '          -1     4 


and    write   the   similar   condition     for    two    lines,   each    defined    by    two 
points. 

4.  Two  conjugate  iina;.';inarv  lines  beiiiL,r  defined  a>  lines  Midi  that 
cadi  contains  the  conjugate  imaidnarv  point  of  an\  point  of  the  other, 
show  thai  if  two  conjugate  iniairinarv  lines  intersect,  the  point  of  inter- 
sect ion  and  the  plane  of  the  two  lines  are  real.  Hence  M'IOW  that 
eon  pirate  ima  vmarv  lines  cannot  be  on  an  ima'_i'inar\  plane. 


200  TH  KKK-  1)1  M  ENSK  )X  AL  (}  KOM  ETHV 

5.  Show  that  if  a  plant'  contains  two  pairs  of  conjugate  imaginary 
points  which  arc  not  on  the  same  straight  line  the  plane  is  real. 

6.  Two  conjugate  imaginary  pianos  being  defined  as  planes  such  that 
each  contains  the  conjugate  imaginary  point  of  any  point  of  the  other, 
show  that  the  plane  coordinates  oi   the  planes  are  conjugate  imaginary 
quantities,  and  conversely.     Prove  that  two  conjugate  imaginary  planes 
intersect  in  a  real  straight  line. 

85.  One-dimensional  extents  of  points.    Consider  the  equations 

F-^/.co, 

PV^CO. 


where  t  is  an  independent  variable  and/.(0  &YV  functions  which 
are  continuous  and  possess  derivatives  of  at  least  the  first  two 
orders.  We  shall  also  assume  that  the  ratios  of  the  four  functions 
J\(t)  are  not  independent  of  t.  Then,  to  any  value  of/  corresponds 
one  or  more  points  j^:  ./•,:  ./•,:  j-4,  and  as  t  varies  these  points  describe 
a  one-dimensional  extent  of  points,  which,  bv  definition,  is  a  <'i/rc<'. 
It  is  evident  that  because  of  the  factor  p  the  form  of  the  functions 
,/',-(  O  imiy  be  varied  without  changing  the  curve,  but  there  is  no 
loss  of  generality  if  we  assume  a  definite  form  foi\/j.(Q  and  take 
p  =  l.  ' 

Let  //,  be  a  point  /'  obtained  bv  putting  /  —  /,  in  (1  ),  and  let  (,> 
be  a  point  obtained  bv  putting  t  —  f,  -(-  A/.  Then  the  coordinates 
of  (t)  are  >/t-\-  A//.,  and  the  points  /'  and  (t>  determine  a  straight  line 
with  the  equations 

PJ',  =  .'/,-+  P  (.'/,-  +  -V,) 

or  err  —  //,  -f-  XA//,,  ('2) 

where  the  ratios  of  A//,  and  not  the  separate  values  of  these  quantities 
are  essential.  As  A/1  approaches  y.ero  the  ratios  A//,:  A//.,:  A/A,  :  A//( 
approach  limiting  ratios  ////,:  ,/>/,:  ,/_//.,:  <!//4=.t'((  tl}:f'.(tl'}:Jl(tl  )'-.<'[(  ^  )- 
and  the  line  (1^)  approaclies  as  a  limit  the  line 

p.r=  //,+  X./y,  .=  f.  (/,)  +  X/;'(/,).  (  :{  ) 

whicli  is  called  the  t<i/i</,'/it  liitf  to  the  curve.  At  /•/•(•/•//  />"/'nf  '//'///*' 
i-n/'i'i-  nt  /r/it'i'/i  tin'  fuiir  ih'l'li'ittil'i'K  /*(  /  )  </'/  imt  ranixli,  t/n'/'i'  IK  it 
</<  flint!'  filni/i'/it  linn. 


POINT   AND   PLANK  COORDINATES 

The  points  //.  and  //,+  '///,.  whicli  suflice  to  fix  the  tangent  line, 
are  often  called  <-<ii(*t'<-uti>'<-  point*  of  the  curve,  hut  the  exact 
meaning  of  this  expression  must  he  taken  trom  the  fore<roin<r 

~  i  r">  o 

discussion. 

We  .shall  now  show  that  tin-  t<in</fnt  litifx  t«  <t  <-urvc  in  Oie  n<-i</h- 
ItnrJiuud  of  a  fixed  point  <>J  tin'  atrrr  _t"nn  it  point  c.rd'/tt  of  dco  dinitit- 
s/'otix,  unli'Kx  in  (In1  nt'lyhliurhoud  <->t  t/tf  j>"/n(  in  tjut'tttmi  the  curct-  in  a 
atntitfht  /t/tf. 

This  follows  in  general  from  the  fact  that  equations  ( -V)  involve 
two  independent  variables  (}  and  X.  To  examine  the  exceptional 
case  we  notice  that  at  least  two  of  the  functions /j(/)  cannot  be 
identically  /ero  if  equations  (  1  )  do  not  represent  a  point.  We 
shall  also  consider  the  neighborhood  ot  a  value  f}  in  \\hich  _/'(/) 
are  one-valued,  and  shall  take/3(0  and/4(f)  as  the  two  functions 

which  do  not  vanish  identically.    We  may  then  place '-f-       —  r  and 
replace  equations  ( 1  )  bv  the  equivalent  equations       '  4 


liei'e  /•'  (  T  )  and  /•',(  T  )  are  one-valued  in  the  neighborhood  considered. 
The  equations  of  the  tangent  line  are  then 


and  the  points  on  these  lines  form  a  two-dimensional  extent  unles 
/•;(T1)+X/-';(r])-(/)/T1-f  X).  (/  .     1,  -J)  (.") 

From  this  follows,  hv  different  iat  ing  (  •>  )  with  respect  to  X, 

/•;'(  rt  )  =  </>;  (r,  -f  X),  (d 

and  hv  differentiating  (•>)  with  respect  to  r}, 

/''/(  T^-f-X  /•',"(  Tj")         (/),'(  T,-f  X),  (7 

and  from  (  <i  )  and  (  7  )  we  have  /•'"(  T,  )     -  d  :   whence  /•'  (  T,  )  -  -•  ,T  +  <•_., 


2tll>  THIiEE-MMENSlUNAL  GEU-METliY 

Equations  (4  )  then  reduce  to 


p.r.       T. 
,,•=1. 

These  are  the  equations  ot  a  straight  line  and  the  theorem  is  proved. 

('nli-idcr    How     three     points,     /',     (t>.     It,     on     till'    dine     (1)     with 

the   coordinates  //,.//,+  A//,  .  and  //1  -f-  A//_  -f  A  (  //,  +  A//_  ).  the   incre- 
nieiiis  corresponding  to  the  iiiei'einent  A/1  ;    that   is, 
;/      /'(/*,).  //,  +  A//,  =.''.(  /j  +  A/),  //,  +  A//,  4-  A(//.  +  A//i  .)=./i(/1  +  'JAO. 
Then  liv  the  theorem  of  the  mean, 

A//  =/,.(/,  +AO-X-('1)=(/iV1)  +  e1)Af, 
and  hv  e\pan>ion  into  Maclaurin's  series. 
\-if=fi(t  -f  J  A/)-  2./)( 


The  three  [mints  /'.  o,  and  It  (letennilie  a  plane  whose  eoordi- 
nates  "  sati>tv  the  thri'e  eijuations 

"i'*i  ^  "-.//..+  "  .//(  "+"  "1//»=  ^' 

".A//.  +  //.A//..4-  "  A//.  -i-  //.A//.      <.).  (s  ) 

//  A"//.  -1-  "  A"//.,  -^  "  A"'//.+  //  A"//4  —  <>. 

As  A'1  approaches  /.ei'o  the  three  points  7',  n.  and  It  approach 
coilieideiict',  and  the  plane  (*)  appi'oaelies  as  a  limit  the  plane 
\\-hose  coordinates  satist'v  the  three  eijnations 


This  plane  is  calli-d  the  <n*i-iiliif!n;i  /-/'///-  at  the  point  /'.  It  is 
evident  that  •"'  <tn</  j>"int  /'  ///./•/  />•  in  <j,  ,/./•<//  <i  <l<linit<  <lffiil<(tin</ 
/>[>i/n.  'I  he  uidv  exceptions  occur  \\ln-n  the  jmim  /'  is  such  that 
the  solution  nf  the  t-ijiiatii  ins  ( 1( )  is  indeterminate.  Writing  these 
c(|Uations  \\ith  derivatives  in  place  «\'  dif't'ereiit  ials  we  have 
n,  t\(  f.  )  -i-  a  t\(  /}  )  -u  //  ;'  <  / .  ,  -u  ,/  t'  i  ; .  |  -  il. 

"}/l    /      )-(-"•'"(/.)    -I-    //,/'(/.)     -t-    '!./.<>      )     =   0.  Cl"    ) 

"./'/;  '.  i   •    <>_•">  '    i   L  //  f.'(  '.  ,  •    ",./*'(  /,  )=  <». 


POINT   AND    H.ANK  COORDINATES 

and  in  order  that  the  solution  of  these  equations  should  he  inde- 
termimuit  it  is  necessary  and  sullicient  that  /  should  satisfv  the 
e([uatioiis  formed  l>v  equating  to  /.ero  all  detenniiuuits  of  the  third 
order  formed  from  the  matrix 


If  these  equations  have  solutions  thev  will  lie  in  general  discrete 
values  of  ti  which  give  discrete  points  on  the  curve  at  which  the 
osculating  piano  is  indeterminate.  To  examine  the  character  of  a 
curve  for  which  the  oscillating  plane  is  evrvwhere  indeterminate, 
it  is  convenient  to  take  the  equations  of  the  curve  in  the  form  (4). 
Equations  (10)  then  take  the  form 

?/,/•',(  T)+  ",/•:,(  r)+  H..T  +  n  t  =  0, 

//,/-','(  T)  +",/''(-)+":,  ()-  (11) 

//,/••,"<  T)  +  >i.,/-"J  (T)  =  0, 
and  these  have  an  indeterminate  solution  when  and  oiilv  when 

A\"(T)=0,      A*'(T)=0.  (1^) 

If  equations  ('.')  are  true  for  all  values  of  r.  the  curve  is  a 
straight  line,  as  has  ahvadv  been  shown. 

Equations  (1")  determine  ?/.  as  functions  of  the  jiarauieter  /f 
Therefore  ///''  nxculcitinii  j>/<iin'><  ';/'  f'  nn'i'f  f<*rin  ///  //encral  n  <>//>•- 
t'linit'nnfon(tl  f.rti'tif  <>f  jit/mi'*.  An  exception  can  occur  oiilv  \\lien 
the  ratios  of  >if  determined  liv  (1")  are  constant.  To  examine  tins 
case  take  again  the  special  form  (  1  )  of  the  equations  of  the  curve 
and  consider  equations  (11).  If  the  ratios  //t  determined  l>y  (11  » 
are  constant,  it  is  lirst  ot  all  necessar\"  that 

/••;'(  T)    -  ,-,/••,"(  T): 

whence  !•',(')       <'J'',(  r  )  -(-  -  <'.,r  +  >'^. 

I'^iiat  ions  (  1  )  then  heci  uue 


TI 1 1! FF   DIM  KNSK  »N A  L  ( i F<  )M  KTK  V 

and   any    point    whose   coordinates   satisfy    these   equations   lies   in 
the  plane  ,       .        .  _  n 

It  is  evident  from  the  dctinition  that  this  plane  is  the  osculating 
plane  at  every  point  of  the  curve,  and  this  can  be  verified  from  equa- 
tions (11).  \Ve  mav  accordingly  make  more  precise  the  theorem 
already  stated  by  saving  that  the  nuculaiinij  planes  i  if  a -curve  in  (lie 
n>/'//J"<r//"»<l  <>t' it  ti.i'i '</  point  nf  the  <-nr>'i- ^tun/i  <t  nne-Jimenxiunal  ejii<nt 
<>f  planes  >//(/.. s'x  the  i-iiri'i-  /x  n  i>Iirni'  r//r/v  in  (lie  neiiihl><>rhnnd  <'<>nxidereil, 

>       /  /  ' 

If  from  equations  (  1  )  the  parameter  t  is  eliminated  in  two  ways, 
there  results  two  equations  of  the  form 

f  ( r  .    ?•  .    ?•  .    ?•   ^  =  0. 

(13) 


Conversely,  any  equations  of  form  ( 1 -V)  may  in  general  be  replaced 
by  equivalent  equations  of  form  (  1  >. 

EXERCISES 

1.    Show  that  in   nonhomogoneous  coordinates  the  equations  of  the 
tangent  line  and  the  osculating  plane  are.  respectively, 

dx  d,j  tlz 

.V  -  .r       }'  -  ,/      Z  —  :: 

and 


2.  Find  the  tangent  line  and  osculating  plane  to  the  following  curves  : 

(  1  )  The  cubic,  jr  —  f:\  >/  =  f:,  :;  =  f. 

I  L'  i  The  helix.  ./•  =  n  cos  6.  //  =  "  sin  9,  ::  =  /.O. 

«(."»)  The  conical  helix.       .r  —  t  cos  /.   if  =  t  sin  /.  ,v  =  /•/. 

3.  Show  that  the  osculating  jtlane   may  be  detincd  as  the   plane  ap- 
proached as  a  limit   by  a  plane  through  the  tangent  line  to  the  curve  at 
a  point  /'  and  through  any  other  point  /''.  as  /''  ajiprnaches  /'. 

4.  Show  that  the  osculating  ]ilane  mav  al>o  be  detined  as  the  plane 
appi-oached  as  a  limit  by  a  plane  through  a  tanp-nt  line  at  /'  and  parallel 
t"  a  tangent  line  at   /'',  the  limit  bciir_r  taken  as  /''  appi'oachc^   /'. 

5.  The  principal  normal  to  a  curve  is  the  line  in  the  osculating  plane 
perpendicular  to  the  tangent  at  the  point  of  contact  :   the  binomial  is  the 
line  perpendicular  to  the  tangent  and  to  the  principal  normal.    Find  the 
equations  of  these  normals. 


POINT    AND    I'l.ANK   rooKDIN  ATKS  'JO.', 

86.  Locus  of  an  equation  in  point  coordinates.  Consider  the 
tMlUilti"n  /(•'-,  jr,,  ./•„,,  ./-,)=<>,  (1, 

where  /'  is  ;i  homogeneous  function  of  ./• ,  ./•„,  ./',,,  and  r,  which  is 
continuous  ;ind  li;is  derivatives  of  at  least  the  first  two  orders. 
Two  of  tin*  ratios  ./^  :./•.,:  .r( :  ./•  can  lit'  assunit'd  arbitrarily,  and  the 
third  determined  from  the  equation.  The  equation  therefore  defines 
a  two-dimensional  extent  of  points  which  l»v  delinition  is  called  a 
surface. 

If  _/'  is  an  algebraic  polynomial  of  decree  //,  the  surfaee  is  called 
a  surface  ot  the  nth  "/•'//'/•.  AHII  xtrtift/Jif  ////»•  nn'i'fx-  </  N?//-M<V  at'  t/i>- 
nth  arili't'  i/i  n  pinnta  <>/•  f/fx  rnfirfly  an  thf  xu /•('<!<•<'.  To  prove  this 
notice  that  a  straight  line  is  represented  by  equations  of  the  form 

PJ\—  ,'/,•+  H< 

\vhere  //,.  and  ^i  are  fixed  points,  and  that  these  values  of  .?;  substi- 
tuted in  (1  )  L^ive  an  equation  of  the  >i{\\  order  in  \  unless  (1)  is 
satisfied  identically. 

A  tiniiji^it  I'm  I-  to  a  surface  is  defined  as  the  limit  line  approached 
bv  the  secant  through  two  points  of  the  surface  as  the  two  points 
approach  coincidence.  Let  _//i  be  the  coordinates  of  a  point  /'  on 
the  sui-face  and  //,  -f-  Ay.  those  of  a  neighboring  point  ft>  also  on  the 
surface.  The  points  /'and  $  determine  a  secant  line,  the  equations 

"f  Wl'ir1'  ;UV  p-.^y.+  X^/.+  Ay,), 

which  can  also  be  written 

P-'\  =  '.I,  +  H^.'/r  (  -  > 

where  the  ratios  of  Ay.  and  not  their  individual  values  are  essential. 
Now  let  the  point  (t>  approach  the  point  /'.  moving  on  the  surface, 
so  t  hat  1  he  rat  ios  Ay  :  Ay, :  Ay.. :  Ay  approach  definite  limiting  rat  ios 
1 1 ;i  :  '///., :  <///., :  </>/  .  Then  t  he  line  (  '_'  )  approaches  t  he  limiting  line 

P->\      '/,  4~  P- '/'/,,  '  •'  * 

which  is  a   tangent    line  to  the  surface  at    the  point    /'. 


'J(Jt;  TIIKKK   DIMKNSIONAL  (  i  K<  >M  KTKY 

Hv   Killer's  theorem  for  homogeneous  functions  \\c  have,  since 

//,  sal  istics  equal  inn  (  1  ). 

rf  ft'  rf  rf 

'/,  +.'/„       '         +   .'/,     -'         +   //4  =    °'  ('I) 

r//,  "  'V/2  '    r//,  '//< 

Hv  virtue  of  (4)  and  (~>)  any  point  .r  of  (3)  satisfies  the  equation 

r/'  rf  rf  rf 


This  is  the  equation  of  a  plane,  and  its  coefficients  depend  only 
upon  the  coordinates  of  /'  anil  not  on  the  ratios  <l>i^:  "'//.,-'  '/,'/.,:  "V/4- 

Hence  all  points  on  all  tangent  lines  to  the  surface  satisfy  the 
equation  (  '»  ).  Equation  (  li  ),  however,  becomes  illusive,  and  the  dis- 
cussion which  led  to  it  is  impossible  when  /'  is  such  a  point  that 

r  f  'c  f  rf  i  f 

=  '»,        '     =  0,        •     =  0,       •     =  0. 

r.;/i  r!'-i  r.;/3  f'H\ 

Points  which  satisfy  these  equations  are  called  shi<pil<ir  j>'>inta, 
and  other  points  aiv  called  r>';i>il<ir  j>»>>if*.  AVe  have,  then,  the 
followiiiLT  theorem  : 


In  the  equation  (  fl  )  the  point  i/;  is  called  the  point  of  tan^encv. 
Converselv.  <//'//  ////<•  i/rnn'ii  in  tin1  tuiii/i'iit  ]>lnni'  tlirniiifli  tin-  /mint 
*  >f  tu  niji  ifij  /x  ii  tiimji'iif  lin>'.  To  prove  this  take  ,?r  any  point 
in  the  plane  ( ti  ).  Then 


and   the  equations  of  tin1   line  through  //,   and   ,rt  are 

P->\  =  iti  +  X~V 

I>ut  a  point  f,>  on  the  surface  may  be  made  to  approach  /'  in 
such  a  way  that  <l;i  :  </>/, :  '/// ,:  •///  :  :  ;, :,:.:,:.  since  the  only 
restriction  on  «///_  is  <_nven  b\-  (1).  which  is  satisfied  by  .:  .  Hence 
the  line  determined  by  _//|  and  :_  lias  e(|iiations  of  the  form  ('•'>)  and 
i-  therefore  a  tangent  line,  and  the  theorem  i-  pro\eil. 


POINT   AND    1'I.ANK   COORDINATES 

The  plane  coordinates  of  the   tangent    plane   to  the   surface  {  1  i 
arc,  from  ( *! ). 

P",  (/  =  !,  2,  :'>.  -1  )          <  7  j 

( '.'/, 

The    coordinates    V,    call    lit-    eliminated    between    these    equal  ions. 

and  the  equation 

/(//,.  //.,,  >/..,  //t )-  0  <  s  > 

found   iiy   substituting    >/,  lor  .r    in  (1).    There   an-   three   pos.sihlc 
results : 

1.   There  may  he  a  single  equation  of  the  form 

(/)(//,    H ,,    a  ,    H    )  --  0  (  ','  ) 

This    is    the   general    ease,    in    which    the    equations   (7)  c;m    be 
solved    and    the    results   suhstituted    in   (  S  ). 
The  condition  for  this  is  that   the  .Jacohian 


ru,      ( ii .,      fit,      iru.,  f't'         f't'         ft'  'f't' 

r.r        f  .r        f  .r        f  .r  c.r,f  r       ft''      cj\r.r,,      r  ./•/./' 

1  '2  4  _  1          -  '    -         4 

f  II ,       fll^       (II,,       fit, I  f't'  f't'          f't'          f't' 

ij'       f  ./\      f.r.,      ( .r  f./y./'.j      ( .1  .f  .1'^      f  .1",       f  .>'.,'  .<\ 

shall  not    vanish.      In    this   case   the    tangent    planes   to   (1)   form   a 
two-dimensional  extent   and  their  coordinates  satisfv  ('.'). 

It  (h  (  a  ,  u  .  >/,.  ?/   )  is  an  algebraic  polvnomial  of  the   ///th  degree, 

1  'J  ."  4  ' 

the  sui'face  (  1  )  is  said   to  be  of  the  ///th  class.      77* /••<?/<///  •/////  Kfr>n';//i 
hiii    at  iiliiiii'K  1'iin  In-  iitiMxi't/,  tiiii'ii'nt  tn  :t  siir/'iii'i   ">'  tin    nit//  i'/<ixx.      I  o 
pro\  e    this    notice    that    a   plane   through    an\    straighl    hue    has   the 

coordinates 

P" ,  ==  '', 4"  ^•'/', • 

where   '-,.  and   /'•,  are   fixed   coordinates.     These   values   o|    nt  substi- 
tuted  in   (  '.'  )  give  an   equat  ion  ot   the  ///t  h  degree   in   \.      I  his  proves 

the    theorem. 

For  example,  consider  the  sui'face 


•JUS 


TIIKKK   m.MKNSlONAI.  <  i  K(  >M  KTR  Y 


Tin-  coordinates  of  its  tangent  plane  are 

P",      ",.'/,' 
and  these  values  sulkst  ii  utcd  in 


a.  ~      »: 
«'ive  '-  +  -'- 

"i         ", 

Tlit'  ordtT  and  class  of  this  surface  arc  l>otli   '2.  but  the  class  of 
a  surface  is  not   in  ^ciieral  equal  to  its  order. 
L*.    There  inav  he  two  etjuations  of  the  foi'in 

4>(  "r  »,.  /',,  ?/4)=  0, 
~^r  (  ii  .   /',.   //  ,,   ti   )  =  0. 

In  this  case  the   tangent   planes  to  (  1  )  form  a  one-dimensional 
extent.     The  surface  is  called  a  i?t'i't'l<>j)<ibl>'  .v///;/'</<v. 
I-'or  example,  consider  the  surface 


The  coi'irtlinates  of  a  tangent  plane  at  //,  are 


P"   —  V  —  V  • 

r     i       .'.i       .'4 

The  elimination  of  _//i  from  these  equations  and  the  equation 


//.-(-  u4—  0, 
//,"'  -f-  "  ,"'  —  "  "     :  '  '. 

;•>.    There   mav   he   three  equations  ol    the   foi'lil 
<^>  (".",,",,//)        '  '. 

•J^  (//,//,      //     ,      //      )        :    <  I. 

Y  f  ".",."..,")     "  "• 

The<e    equatiniis    can    lie   solvcil    In]-    //  .     Ileiiec    in    tins   ea.M>   the 
tangent  planes  tdnn  a  disci'ete  system. 
|-"or  example,  consider  the  sin-face 


POINT   AND    PI.ANK   COORDINATES 

The  tangent  planes  have  tin-  coordinates 


These   lead   to  the  equation 


Tin1  tangent  planes  are  the  two  planes./-  ---(land  ./•  +./•„-(-./•.-  <>. 
In  fact  the  surface  consists  of  these  two  planes. 

EXERCISES 

1.  Show  that   the  section  of  a  surface  made  l>v  a  tangent   plane  is  ;i 
curve  \shich  has  a  singular  point  at  the  jioiiit,  of  contact  of  the  plane. 

2.  Show  that   Ihe  section  of  a  surface  of  the  n\  h  order   made   \\\  any 
jilaiie  is  a  curve  of  the  /;t  h  ordei1. 

3.  Show  that   anv  tangent    plane  to  a   surface  of  second   order   inter- 
sects the  surface  in  t  wo  straight  lines,  and  in  particular  that  the  tangent 
plane  to  a  sphere  intersects  the  sphere  in  two  minimum  lines. 

4.  Show  that    through   the    point    of  contact    of  a   surface  and  a   tan- 
gent  plane  there  -'o  in  general   two  lines  Ivinc,  in  the  plane  and   having 
three  coincident   jmints  in  common  with  the  surface. 

5.  .Show  that    the   equation  y't,'',  >•'*.,••''.)       *'•  \vhere   the    function    /'is 
homogeneous    in   ./•  ,  ./'.,.  ,»'    and   the  coi'irdinatc  ./•     i.~,    IIII.-.SIIIL,',    represents 
a    cone,    liv    showing    that    it    is    the    locus    of    lines    through    the    point 
0:0:0:  1. 

(>.  Show  thai  the  tangent  plane  jo  a  cone  contains  the  element  of 
Ihe  cone  through  the  jioint  of  contact. 

7.  l''rnm  l']\.  ,"»  slnt\v  that  in  nonhomopMieoiis  Cartesian  coordinates 
the  eijiiaiion  1\.i\  //.'.)  -  0.  where  /'  is  homogeneous,  represents  a  cone 
with  its  vertex  at  the  oi'i^in  and  1  hat  _/'i ./',  //'  '*  represent  s  a  cvlinder 
wil  h  its  elements  parallel  to  n'/.. 

S.  Show  that  through  a  singular  point  of  a  surface  there  j,roe-s  in 
general  a  cone  (if  lines  each  nf  \\hich  has  three  coincident  points  in 
com  mi  m  with  the  surface. 


210  T1IRFF   DIMENSIONAL  OFo.MKTRY 

9.    Find   the  equation   or  equations  satisfied  by  the  coordinates  of 
the  tangent   planes  of  each  of  the  following  surfaces: 
,  1  )   '_'  u.rrr,  +  Ar/f  +  07  -  0, 
-f  /,.r.r  +  r./Y  =  0, 

•;-  +  '-.r|  =  o. 


10.  Sh«\v   that    tli.-   tangent    planes  of  a   cone  or  a  cylinder   form  a 
one-dimensional  extent. 

11.  It'  the  fi[uation  of  a  surface  is  written  in  the   mmhomogenoous 
form   '.  =  f\.i\  >n,  show  that  its  tangent  planes  form  a  two-dimensional 


1:2.  Show  that  two  simultaneous  equations  $  (./•  .  ,r,.  ,r.,,  ,r  i  =  0  and 
<5  i./-..  .r,,  ./-;.  ,;•,  i  =  (I  define  a  curve,  and  that  if  the  tangent  planes  to 
the  curve  are  defined  as  the  planes  through  the  tangent  lines  to  the 
curve,  thev  form  a  two-dimensional  extent  Driven  bv  the  equations 

nil   -          1  —  A    '"    toLretlu'l1   with   the   equations   of   the   curve. 
r.r,          f  ./•• 

87.   One-dimensional  extents  of  planes.    Consider  the  equations 

*"',=/,(  0. 

P"    ---  i'  ( f ), 

•  ;  (  1  ) 

?".,=/:,<  0, 


where  ?/,  are  plane  foJ'inlinates,  /  an  independent  variable,  and 
f.(t)  functions  of  /  which  are  con- 
tinuous and  possess  derivatives  of 
at  least  the  lirst  two  orders.  \Ve 
shall  also  assume  that  the  ratios  of 
the  t'i  nir  fiinct  ions_r  <  f  )  are  in  it  in- 

(!*'[  iclldeilt  i  'f  /.      The  ciplat  i<  >11S  1  hell 

define  a  one-dimensional  extent   of 

planes.     1.,-t    /•    be    the   coordinates 

of  a  plane  /<  (  Fi'_r.  •>'•'<)  obtained  bv 

j>laciii'j  t    -1    in  C 1  )  and  let  '•,. +A'-t 

be    the    eoordinates    (,(     ;i    plane    </ 

found   by  placiiiLT  f-    ^-(-A/.     Then    /•   and  7  determine   a   straight 

line    in.   the  equal  ions   >  >!'    which    are 

P"        '•   4-  AM  '•   -f-A/1,  ) 

'  il1  (TH       -    '•      -    \A''  . 


POINT   AND    I'LANK  C'OOllDl.  \ATK.S 


211 


As  A/1  approaches  /cm  tin-  line  m  approaches,  a  limitiii'j;  liiu-  /, 
of  which  the  equations  are 

pn.=  •',  +  X./r.  =>/;.(/j)+  V','^,)-  (-) 

This  line  is  called  a  rluirnrtt'i'txtit'  ol  tin-  extent  defined  li\-  (  1  ). 
It  is  evident  that  ///  <ini/  jiliim'  <>t'  tin-  f.iii'iit  f»r  <rlii<-li  tli,-j'<mr  di-ric- 
atif't  \s  f\  (  t  )  </"  nut  VilHtK/t  t/tfi-i'  t*  <i  <l>  finite  (.'/m/'ilct  i-  fistic. 

\Vf  shall  iu»w  pr<>\  e  the  pn>])ositioii 

The  t'luinii'ti'i'isticn  form  m  //»•///  /•»//  <f  *///_•/  '(/.•«•  /<*  ^7//,//  ,</,•//  /il/im'  "f 
tin1  dt'fininy  Jjlitne  f.rtint  t*  t<in</iut  ///»/i</  tin-  i-ntii'i'  f/it(/'iii-(i-rixtic  i/t 
th'tt  j>/'t/(C. 

To  pi-ovc  this  we  notice  that  anv  point  ./-(  which  lies  in  a  char- 
acteristic satisties  the  two  equations 

2*  -''./'/      •'••  *        j  >f    -  ( 


and  that  in  general  /  iua\p  he  L'liininati'd  t'roin  thoe  t-ijitations  with 

a  result  of  the  form 

^(/•j,  j-a,  J'a,  ./;,)-    '».  (  }  ) 

This  proves  that   anv  point   on  anv  characteristic  lies  on  the  sur- 
face1 with  the  ei|uat  ion  (  4  ). 

l»y  virtue  of  the  manner  in  which  (1  )  was  derived  we  mav  \\riti- 


Avhere  /  is  to  lie  deterinincd  as  a  function  oi   ./•_  troin  the  second  o! 
equations  (  :>  ).    Thercfi  >re 


This  shows  that  the  tangent  plane  ot  (  1  )  is  the  plane  >/  o!  the 
extent  (  1  )  and  that  the  same  tangent  plain'  i^  found  for  all  points 
for  which  f  has  the  same  value:  that  is,  for  all  points  on  the  same 
characteristic.  The  proposition  is  then  proved. 

Consider  now  three  planes,  <v,  /\ -f-A'v,  /v -f- A>v  +  A( '-(  rA-1). 
1  hev  detei'inine  a  point  /'  the  coordinates  ot  \\hieh  satistv  the 
three  eijiiations 


A-'/'    '- 


212 


TIIK  K  K-  DIMENSIONAL  ( i  K( )  M  KT  R  Y 


and    as   A/    approaches   /.fro    the    point    /'   approaches   as   a   limit    a 
point    L   tin-   coordinates   of   which  satisfy   the   equations 


(0) 

V       ' 


./•</'-> 


./•  ,/-/•  —  o, 


or,  what  is  the  same  thiii!^'.  the  equations 

•'•,.M  0  +  •'•,•/_•(  '  >  +  .';,>;(  f  )  +  JrJ\(  0  =  0, 

vV)  +  -'-,/^o  +  •'•;;/;'  (')  +  .'•,/.;(/)  =o,  <?> 

./•,/','(     /     )      +     l'.Jr"(     f    )     +     ./;,/';'(      t     )     +      J'J'"(     f     )     -      0. 

I'lie  point    /,  we  shall   call   the  limit  point   in   the  plane  >\  and  shall 
prove  the   following  proposition  : 

77/.  •  JIK-IIX  i>f'  tlif  limit  fiu'iiits  /x  in  i/i'tiiTiil  it  I'i/rri,  <-<tlli'il  ///<• 
ri/xj'iiltll  i-i/i/i',  t"  H'Jiifli  tin'  i-)i«  rtti-tt'rixti<-x  iii','  til  ii'/i'iit  . 

The  lirsi  part  of  the  proposition  follows  from  the  fact  that  equa- 
tions (f>)  can  in  general  he  solved  for  ./-,  as  functions  of  /. 

To  prove  the  second  part  of  the  proposition  note  that  hv  differ- 
entiating the  first  two  equations  of  (7)  on  the  hypothesis  that 
./•j,  ./•„,  ./•„,  ./'(  and  /  vary,  and  reducing  the  results  hv  aid  of  the 
three  eqiiat  i<  ms  (7  ),  we  have 


and  t  I'om  <  7  )  and  (  <s  )  anv  values  ot  the  coordinates  .V  \\'hieh  sat  isfv 
these  cipiat  K  ins  sat  1st  v  alsi  » 


that   is.  the  point   .\\  lies  on  the  characteristic  (  •>  ). 

To    complete    the    general    discussion    we    shall     now    prove    the 
[  >r<  ipi  )>it  ion 

77/.'      "Xi'Ufiftllli/     /'/'Itn.--     a/    f/,,'     r/lsfi/i/il/    ,</•/,'     Hi',-    til,'    filililiH     <>f    tin 
ill  t:  ii  1  1/'/     ['III  in      i  J't  flit  . 

\\\    different  iat  in^   the    lirst    of   equations    (7)    and    reduciii'41   hv 
the  aid   of   the  second  equation,  we  have  Wj^/j.f  /  )       ".      Therefore 


POINT   AM)    PLANK  COORDINATES  213 

by  selecting  tin-  proper  equations  from  (  '>  )  and  (  s  )  and   replacing 
/•(  O  bv  '•_,  we  have  the  conations 

y  »•/,=  »,   v,v/.,-^u.   y  <-.,/-./•,=  o. 

I»ut    from    (I'),    ^  <s->,    these    equations   deline    rt   as    tin-    osculating 
plant-   nf   the   cuspidal    rd^v.     This   proves   the    proposition. 

In  the  foregoing  discussion  \ve  have  considered  what  happens  in 
general.  To  exaniine  the  exceptional  cases  we  niav,  as  in  sj  S;">, 
write  the  eiiuations  (  1  )  in  the  form 


The  eijuatioiis  (  '•}  )  for  the  characteristics  no\v  becom 


J',  /••,'(  T)  +  ^,  /'',!(  T  )  +  ./',  =0, 

and  tin;  ecjuations  (7)  for  the  limit  points  become 
./-,  /•',(  T)  +  j\,  /•'(  T  )  +  .';,T  +  ./4  =  i  ), 

.rlJ'[(T)  +  .rJ-"(T)  +  .r:i  =  0,  (11) 

^,  /•",'(  T  )  +  •'•,/•"'(-)  =0. 

'I  he  second  o|  the  equations  (10)  can  be  solved  for  r  unless 

l'\(r)    --  »•,,  /''.!(  T)=  '•,: 

whence  /•'(  ~  )  —  '',T  -f  e.,  /•'_,(  T)  =  c_,T-f  <-4, 

and  /"I'(T)=  I',  /••;,'(  r)=  0. 

In  this  case  etjuations  (  I'l  )  become 


so  t  hat  all  characteristics  are  t  he  same  straight  line.    At  the  .same  t  ime 
e<|iiat  n  ins  (  '.'  )  beci  une  .       i     . 


P 


" 


'214  T  1  1  R  JO  lv  DIM  K  N  S  It  )N  A  L  (  ',  KOM  ET  R  Y 

axis  <>f  the  pencil  is  the  straight  line  (1~)  with  which  the  charac- 
teristics coincide. 

Turning  now  to  equations  (11)  \v(>  see  that  the  last  one  deter- 
mines -/•:./•„  and  the  others  determine  ./•.,  and  ./4,  unless  /•'"  (T)  =  0 
and  /•'.','(  T)  0.  This  is  the  same  exceptional  case  just  considered. 
The  equations  for  the  limit  points  become  equations  (12),  so  that 
the  limit  point  in  each  plane  is  indeterminate  but  lies  on  the  axis 
of  the  pencil  of  planes. 

Another  exceptional  case  appears  here  also  when  the  solutions 
of  (  11  )  do  not  involve  r.  This  happens  when 

^a"(r)  =  <•',<<>); 

whence  /•',(  r)  =  <\J'\(r)  +  C.,T  +  ca. 

Equations  (11)  then  have  the  solution 

J'l:jY.j'3:j-i  =  cl:-l:cz:c3.  (13) 

At  the  same  time  equations  (9)  are 


=  T, 


All  planes  which  satisfy  these  equations  pass  through  the  point  (13). 

The  surface  of  the  characteristics  is  in  this  case  a  <-<>n<\  since  it 
is  made  up  of  lines  through  a  common  point.  The  cuspidal  edge 
reduces  to  the  vertex  of  the  cone. 

In  vj  si!  we  have  shown  that  the  tangent  planes  to  a  surface 
may,  under  certain  conditions,  form  a  one-dimensional  extent  of 
planes,  and  have  called  such  surfaces  dwclopable  xiirt\t<-fx.  \Ve  may 
now  state  the  following  theorem,  which  is  in  a  sense  the  converse 
ot  the  ;ib(  ive  : 


t'tifji  /><>//tf,  On'  viirfiifi-  //t<i//  In'  <>iit'  nf  thf  1'iilloii'iii'i  thrc?  Icini  /,v  : 

1.    It  nut  *t  f'i    i-'inijHiMftl  i>t'  titm/i'iit  li/ti'x  f»  n  x/nii-i'  ctiri'f. 

'2.     It    until    Li-    it    ''"in.     (If   (/a'    i'>'rti:<'    /N    <tt    i/itinili/.    tin-    <-i>/it'   iff    a 

I'l/JimliT.  ) 
'•'>.    If  mnii  ilt-i/i'/tifiifi'  tntn  t/ti'  nj'ix  (//'</  /ii'itcil  <>t  jthtm'is. 


I'nlNT   AND   PLANE  COORDINATES 


In  the  above  theorem  the  nature  of  the  surface  has  been  de- 
scribed oiilv  for  each  portion  of  it,  since  the  foregoing  discussion 
is  based  on  the  nature  of  the  functions  J\(t)  in  the  neighborhood 
of  a  value  of  /,  which  fixes  a  detinite  plane,  a  detinite  character- 
istic, and  a  detinite  point  on  the  cuspidal  edge.  In  the  simplest 
ease  the  developable  surface  will  have  throughout  one  of  the 
forms  given  above.  Next  in  simplicity  would  be  the  case  in  which 
the  surface  is  composed  of  two  or  more  surfaces,  each  of  which  is 
one  of  the  above  kinds.  It  is  of  course  possible  to  define  surfaces 
which  have  different  natures  in  different  portions,  but  the  char- 
acter of  each  portion  must  be  as  above  if  the  functions/]  (t)  satisfv 
the  conditions  given. 

The  planes  of  the  extent  are  said  in  each  case  to  r/^vAy/  the 
developable  surface. 

88.  Locus  of  an  equation  in  plane  coordinates.  Consider  an 
equation  f^  ^  u ^  (> ^  u^)=  0,  (  1  j 

where/' is  a  homogeneous  function  of  the  plane  coordinates  «,-.    \Ve 
shall  consider  only  functions  which  are  continuous  and  have  deriva- 
tives of  at  least  the  first  two  orders.    Two  of  the  ratios  //   :  </ , :  ;/ , :  // 
can  be  assumed  arbitrarily,  and  the  third  determined  from  the  equa- 
tion.   Hence  ///c  I'l^mitinn  /vy/v.-v///*  an  f.r/,//f  <>t'  tiro  tli/iH'mtivity. 

If/'  is  a  polynomial  of  the  //th  degree,  then  n  planes  belonging  to 
the  extent  (1  )  pass  through  any  general  line  in  space.  The  proof 
is  as  in  ^  Mi.  In  this  case  the  extent  is 
said  to  be  of  the  n\.\\  class. 

\Ve   shall    not    restrict    ourselves,   how- 
ever, to  polynomials  in  the  following  dis-          / 
eussion,  but  shall  proceed  to  find  some  of 
the  general    properties   of  the  extent    (  1  ). 

Let    rt   be  the  coordinates  of   a    plane  // 
(I'ig.  .>4)  of  the  configuration  defined  by 
(  1  ).  and  i\  -4-  A'\  those  of  another  plane  y, 
also  of    the   configuration.     The  two  planes  /<  and    </  determine   a 
line  in  whose  conations  in   plane  coordinates  (theorem  I.  ^  <s  \  )  are 


or,   otherwise  \\ritten.        an     •.  r  -f-  /^ 
whel'e    the    ratios   oiil\    of   A''    are   esse 


216  THREE-DIMENSIONAL  GEOMETRY 

Now  let  <i  approach  coincidence  with  p  in  such  a  \vav  that  the 
ratios  A'1  :  A'1,:  A'',:  A''4  approach  limiting  ratios  </>•  :  »/;'o :  »//•  :  «//•  . 
The  line  ///  approaches  a  limiting  line  A  whose  equations  in  plane 
coordinates  are  <n/.  =  *•,.+ /Wr,.. 

'1'he  dilYerentials  (/<•  are  hound  only  by  the  condition 

cf,         cfi         cf-i         cf  , 

,1t  =     -Ji'+    '-  Jr  +    •    Jv  +  -•-</<•  =  0,  (-2) 

rr4 

so  that  the  planes  with  coordinates  ilt^:  <lr^:<lc,:  ilt-  form  a  linear 
one-dimensional  extent  which  by  theorem  II,  >j  84,  consists  of  all 
planes  through  the  point  /',  whose  coordinates  are 

cf     cf     cf     cf 


This  point  lies  in  the  plane  i\  since,  by  Killer's  theorem  for 
homogeneous  functions. 

cf  cf  cf  < f  _ 

1 cc          -  cv          *  f r          4  < c 

1234 

which  is  the  condition  ( 1  ),  ^  <s4,  for  united  position. 

A  line  L  is  the  intersection  of  any  one  of  the  planes  <lc^.  <!(•„ :  Jr., :  <lr^ 
with  the  plane  ^  :  r^.  r., :  r4>  Hence  the  lines  L  form  a  pencil  of 
lines  through  /'. 

The  point  /'  is  not  determined  by  equations  ( ;> )  if 

cf  (  f  c  f  c  f 

'    =  o.      •    =  o,      •   =  u,      •    =  0.  (,-/) 

( r  f r  cv  cr 

i  •_•  3  -t 

A   plane  for  which   these  conditions   is  met   is  called  a  isin</u!<tr 
plnnt-  of  the  extent  (1).     Other  planes  are  called  rct/ulur  ithint'ts. 
\Ve  sum  up  our  results  in  the  following  theorem: 

In  nn  if  r><ja1nr  plnn-  p  of  tin*  c./i,  /if  (  1  )  tin  r>    //.  *  </  <1>  Units  point  /' 

<//M/  hie'  "1  f/t>   j<,  /i<-tl  i/'ith  tlif  rsrtfj-  I'  mt'1  in  t/n'  jilnitf  j»  /,*  t/n'  limit 
"_t  the  utft  /•.•«•'  I'tfiH  "t  i*  iiit<l  n  nt'(i/fJii>/'i/('/  I'l'tiii . 

The  point  /'  may  be  called  the  ///////  point  in  the  plane  p. 

'I  he  elimination  of  -•  from  equations  ( :>  >  and  equation  (  1  ),  written 
in  i\,  will  giy  tin-  locus  of  the  points  /'.  There  are  three  cases: 

I.  Tin-  elimimttion  may  !_five  one  and  onlv  one  equation  of  the 
form  i.  /i:  v 


POINT  AND   PLANE  COORDINATES  217 

The  locus  of  f>  is  then  ;i  surface.     If  the  extent   (  1  )   is  <if  the   //th 
class,  the  surface  ('!)  is  also  called  a  surface,  of  the  /ah  class. 
II.   There  may  he  two  equations  of  the  form 


The  locus  of  /'  is  then  a  ciir\e. 

III.  '1  here  may  he  three  equations  connecting  .r},  r,,  ./-(,  j-  .  The 
points  /'  are  then  discrete  points. 

We  shall  now  show  that  the  planes  of  (1)  are  tangent  to  the 
locus  of  /'  in  such  a  manner  that  /'  is  the  point  of  tangencv  of 
the  plane  j>,  in  which  it  lies. 

To  prove  this  write  equation  (4)  in  the  form 


and  differentiate.     We  have 

Yr,/.r  - 

^   i      , 

which,  1>\'  aid  of  (  '2  )  and   (•'!),  is 


Consider  now  in  order  the  previous  cases. 

I.  If  .r.  satisfy  a  single  eijiiatioii  (  t!  ),  we  have 

f  d>    ,          if  d>    ,          (  d>    , 

^   ,ls+    ^  </.,;,+    *  ilj' 
ic'.r  (  ./'  f  .r  c.f 

1  -J  '.',  4 

r>v   comparison  of  (S)  and   (  '.'  )    we   have  p<\~  --'  which    shows 

(  .i\ 

that   '\  arc  the  coordinates  of  the  tangent    to  (/>     -  <>  at  the  point  .r. 

II.  If  ,/•,.  satisfy  the  t\\o  ecjuations  (7),  we  liave 


A    coinjiarisoii    with   (S)   gives  p>\  '  -f  X       '  -   which  sliows  that 

f  ./'  (  .r 

>':  passes  through  the  line  ot  intersection  ot  the  tangent  planes  to 
(/)  'I  and  0,  -  0  and  hence  is  tangent  to  the  curve  delined  1>\  the 
t  \\'o  sii  rtaces. 

III.    II    the   points   ./-(    are   disci'ete    points,    \\c   mav  sa\    that    each 
plane  of  the  extent    is  tangent   to  the  point,  through  \\hich  il  paocs. 


•_!  i  s  TIII;  i-:  i  :  DIM  K  N  s  i  <  >  .\  A  L  (  ;  K<  >  M  KT  i  ;  \' 

thus  extending  the  use  df  tin-  word  "tangent  "  in  a  nmnner  which 
will  he  u>eful   later.    Summing  up,  \ve  say  : 

.1  t  ir,,-,liun  Ksi"iKil  i.rt-iit  "f  jilitiU'X  '•"//.v/x/'x  ,,f  f>!iin>  x  ?/•///'•//  <//Y 
t'lii'i-  nf  .////-•/•  f"  <f  xiir/'i'-f  "/'  /"  ''  'V//w  «/•  />/  ^  j»>inf. 

The  theorem  has  reference,  ot  course,  only  to  the  neighborhood 
ot  ;i  plane  of  the  extent.  The  entire  extent  nuiv  have  the  same 
nature  throughout  or  ilitTereiit  natures  in  different  portions. 

89.  Change  of  coordinates.  A  tetrahedron  of  reference  and  a  set 
of  coordinates  ./•  having  been  chosen,  consider  anv  four  planes  not 
meeting  in  a  point  the  equations  of  which  are 


/    ./•  4.  ,/    ./•  =  i), 

i.;    .;  n    4 


(1) 


il      r   4_  ,/     _/•   4-  ,i     ./•    4-  ,(     ./•    : 
41    1  i-j    -2    '       4.;    :;  41    4 


the  eoeth'cients  bein^  suhject  to  the  single  condition  that  their  deter- 
minant '/ti  shall  not  vanish.  We  assert  that  if  we  place 

then  ./•'  are  the  coordinates  ot  the  point  ,/-(.  referred  to  the  tetrahedron 
formed  h\-  the  four  planes  (1).  '1  he  proof  runs  along  the  same 
lines  as  that  of  the  corresponding  theorem  in  the  plane  (  >j  -V  )  and 
will  acei  ii  diii'_;l\  in  a  he  given. 

It  is  also  e;isv  to  show  that  bv  the  same  change  of  the  tetrahedron 
of  reference,  the  coordinates  >/:  become  n[,  where 

'I  he  change  trom  one  set  ot  ('artesian  eooidinates  to  another  is 
et'fected  bv  means  of  formulas  which  are  special  cases  of  ('_').  If 
(  j".  if  \  z\  t)  are  rectangular  ('artesian  coordinates  and 

,,  ,•  _L  /,  ,/  _i_  ,.  ••  ^_  ,.  /  —  1 1 
r          i  •          i "         i 

,/  ./  -  /;,//  4-  -•  ,:•  4-  -   /  .     ii.  (.{) 

.1      .1  •     —      Li/     4-      ,-       ',:     _U      ,         /      ;       .      I) 

ai'e  an\'  three  iioiii  larallel  nlancs.  and  we  place 


pi/          /'    I  II  J-  4-    /,    ,/  4-   ,-    :•  4-   ,     /   |. 

p?1      L  (  ,i  ./•    '-  I.  ii   -f  <•  :  4-  ,   /  ,. 

pt'    /. 


POINT   AND    PI.ANK   ConKDINATKS  '2\() 

tin1  quantities  ./•',  //'.  ;•',  t'  are  proportional  to  the  perpendiculars  on 
the  three  [danes,  and  it  is  po>xil»le  to  adjust  the  factors  /•_  so  that 
''://':,;•':/'  niav  lie  exact  Iv  the  ('artesian  coordinates  referred  to  the 
jilanes  (4)  as  coordinate  planes,  the  coordinates  liein^  rectangular 
or  ohliijiie  according  to  the  relative  position  ot  the  planes  (  1  ). 

The  equations  (•>)  represent  a  change  from  a  rectangular  set  of 
coordinates  to  another  set  which  niav  or  niav  not  l>e  rectangular, 
and  conversely.  A  change  from  an  oliliipie  svstein  to  another  is 
represented  l>v  formulas  ot  the  same  tvjie,  since  the  change  niav 
he  brought  about  as  the  result  of  two  transformations  of  this  tvpe. 

EXERCISES 

1.  Kind  the  characteristics,  characteristic  surface,  and  cu.-pidal  cdu'e 
of  each  of  the  following  extent   of  plane-;  : 

1 1 1  Ptf}=  1.  pn.,=  :>  t.  pit..= .".  /'-',  P"(  =  /;. 

(L'l    pH{--    -   "I:    Sill    /.    pit,,:     .        -    "/.'    COS   /.    ,J/Y.r    :    ,/'".    f>.',  ,/7.'/. 

('.]}  pil^  \  —  /-.  pit.,--  -  -  /.  pi'..  =  —  I  1  +  f-\,  p",      1  -f  /-. 

(  I  )  p»^   :  '1  t.  pn.,=  f'~—  1.  pna—  /-'  f  1.  pn4        1. 

2.  If  a  minimum  de\  elopahle  is  defmed  as  a  one-dimensional  extent 
of  minimum  planes,  show  thai  the  characteristics  are  minimum  lines  and 
the  cuspidal  edure  is  a  minimum  curve  unless  the  developahle  is  a  cone. 

3.  Show  that    the  necessarv  and  siillieieiit   condition  that    the  surface 

0. 


4.  I'mve  that  planes  which  are  tangent  at  the  same  tune  to  t  u  o 
^riveii  surfaces,  two  LTiven  curves,  or  a  iriven  suriace  and  a  ^iveii  curve 
define  developahle  surfaces. 

f).  l-'ind  the  envelope  of  each  of  the  following  oiie-diiuetisional  exlei  t 
of  plane,; 

(1)2  //,  -r  -  »:;+   I  ".;       L' 1  "f       <>. 

C2)     .".    ",",";        -     ".'^    -    <>. 

I'.'i)  '/,J+  »;    -  "f      ". 

6.  Show    that    the    minimum    planes    form   a   t  \vo-diineiiMoiial   extent 
a.  n  d    find    its    equation. 

7.  Show  that  p.f  {  --  -- j\<  I  i  -f  >;/',  i  /  >  <  1 .-.-"-.    I  i  il«-iine«.       ile\clop:tMe 
surface   and.   eonverselv,   that    an\    dc\elopahle    siii't'ace    u  Inch    i-    ii"1    a 
coi  it'  or  the  axis  of  a  peneil  "f  plane-,  may  he  ex  preyed  in  this  way. 


CHAPTER  XIII 

SURFACES  OF  SECOND  ORDER  AND  OF  SECOND  CLASS 
90.  Surfaces  of  second  order.    Consider  th<>  ('([nation 

5/WV=0'          (."*,=  ",*)  (!) 

which  defines  a  surface  of  second  order  (  ^  M'» ).     The  .Jacobian  of 
£  S»>  becomes,  except  for  a  factor  2,  the  determinant 


called  the  tlim'riminnnt  of  the  equation.    \Ve  mav  make  the  follow- 
ing preliminary  classitication  : 

I.  A  —  0.  The  surface  has  a  doubly  infinite  set  of  tangent  planes. 
The  plane  equation  of  the  surface  mav  be  found  by  eliminating  //, 
from  the  e([itations 

nil    =  n    .r  4-  it    .r   4-  u    .r   4-  <t    .r  , 

'      i  n     i    '        rj    j  i:;    :;    '        14    4 

pi/   =  n    .r  4-  it    ./'  4-  (i    .r  4-  </    .r  . 

'-    '  -4    4  (2) 

P>'..=  '<    •''  -+-  ".,„.''.,  4-  ".,.''.,+  ''..  .''  . 

nil    =  n    ./•   4-  </    ./'   4-  n     ./'   4-  "    ./'  . 

"4  14     1  '.'4     -  :;i     -.  II      I 

and  (M  |  nation  (1  ).     I  Jut  \\  com  bin  at  ion  of  (  '1 )  and  (1  )  L,ri\"cs  readilv 

)i  ./•   4-  u  ,./\4-  //,./•,+  //  ./•   =  0, 
and  the  elimination  of  .r   from  this  ei|uation   and  the  set   ('2)  i_;'ivcs 


SURFACES  OF  SECOND  ORDER  AND  SECOND  (LASS  2'2\ 

This  is  an  equation  of  the  second  decree  in  >/,.  lleiiee  (/  *;//•- 
J\iff  <>f  tin'  yt-ro/iif  'irilt-r  for  //•///<•//  ///«•  tlisfriminant  ?'x  not  zi-ro  /.s  ulan 
<i  xitrt'tiff  of  tin'  xt'/'o/iif  ,'lnxx  (  xj  SS). 

It  is  not  difficult  to  show  th;it  the  discriminant  of  (  •>  )  is  not 
equal  to  /.cro. 

II.  A=l>.  The  tangent  planes  either  form  a  one-dimensional 
extent  of  planes  or  consist  of  discrete  planes.  These  cases  will  he 
examined  later. 

91.  Singular  points.  l>y  vj  Sti  singular  points  on  the  surface  (1  ), 
J;  I'l),  arc  <jiven  by  the  e()iiations 

<i   ./•  -f-  '/   ./•  4-  a   .r  -+-  a   .r  —  0. 

11      1      '  !•_•     J      '  13     S     '  14     4 

(I    .r  +  it    .r   -f-  ''    •''  4  "    .'•  —  (\ 

(  1  ) 
«    .)'  4-  ''    .r  +  "    ./'  +  <t    .>'  —  0, 

l:t     1     '        '.'3    a    '        :i:i    :i  •'.!     -t 

'/    ./•  4-  it    .r  4-  ''    .''  4-  "    .''  =  ". 
u    i          -J4    •:          ;i4    3          ti    4 

There  are  four  eases  : 

I.  A   •'-  'I.     Ivpiiitions  (1  )  have  no  solution,  and  the  surface  has 
no  singular  points.    This  is  the  general  case. 

II.  A=  --  0,  hut   not   all   its  first  minors  are  y.ero.     The  surface  has 
one  and  onlv  one  singular  point.     Let  //.  he  the  coi'irdinates  of  the 
singular  j>oint    and  ,;,  the  eoi'trdinates  of  any  other  point   in  space, 
and  consider  the  straight   line 

pj\=  //.-f-  \r(.  (  •_'  ) 

To  iind  the  points  in  \\-hich  the  line  (  '2  )  meets  the  surface  sub- 
stitute in  equation  (1  ),  £  '.to.  Since  the  coordinates  //,  satisf\-  the 
equation  of  the  surface  and  also  the  equations  (  1  ),  the  result  is 


'I'his  shows  that  anv  line  throu^li  a  singular  point  meets  the  >ur- 
faee  onlv  at  that  point  (  X  =  0  ),  and  there  \\ith  a  doiihlv  counted 
point  ol  intersection.  An  exception  occurs  when  .:,  is  taken  on 
the  surface.  Then  equation  (  ;>>  )  is  identically  satisfied,  and  the 
line  >!'•:  lies  entirely  on  the  surtaee.  I  fence  ///«•  xinfiii-i-  t*  <t  <•"/<• 
iritJi  tin'  xi  in/ill,!  r  jiuiiit  nx  tin'  ri't'fi'.i:  '1  here  is  no  plane  equa- 
tion of  the  surface.  Ill  fact  the  tangent  plane>  form  a  siiiLrl\ 
infinite  extent  of  [dancs,  and  their  coi'irdinates  are  suhject  to  two 
e<  mdit  K  nis. 


2'2'2  TIIKKIv  DIMENSIONAL  (iKOMKTKY 

III.  A     ".  all  its  first  minors  are  y.ero,  but  not  all  its  second  minors 
arc  /.t>ro.    Hi  [nations  (  1  )  contain  two  and  only  t\vo  independent  equa- 
tions and  hence  the  surface  lias  a  line  of  singular  points.     If  tins 
line  is  taken  as  the  line  .1^=  <>.  ./•„  =  0  in  the  coordinate  system,  equa- 
tions (  1  >  show  that  we  shall  have  <i    =<t    =  it,,.  =  <i  ,  —  <>...=  ".,  =<i    =  '">. 
and  the  equation  of  the  surtace  becomes  "u.r~  +  '2  «/)  .,.'•,./•.,+  ''.,„•''.;'  =  "• 
At    least   two  of  the  coefficients  in   the  last  equation  cannot  vanish, 
since   the   surface   has   only   the   line  ./•  =0  and  r,=  0   of  singular 
points.    Therefore  the  left-hand  member  of  the  equation  of  the  sur- 
face  factors   into  two   linear   factors.     Hence  tin'  .<>•///;/'//<••'  i-nnaiatx  ,f 
fir,,  ,//>•////-•/  I,I,IH,'X  inti'rxi'i-f/'iii/  in  tin'  Jim'  i  >f  sitHjuldr  point*. 

IV.  A-:-",  all   its   tirst   and   second   minors  arc  /.ero,  hut    not    all 
the  third  minors  are  /.ero.     Equations  (  1  )  contain  one  and  onlv  one 
independent    equation,  and   hence   the   surface   has  a   plane   of  sin- 
gular points.     If  this  plane   is  taken  as  ./-,  =  0.  the  equation   of  the 

surface    becomes    ./••=-<>.      Hence    tin'    vnrf'i'-i'    i-o/tx/sftt    <>f  tl«     ^Lnn-    »f 
xiinjul'ir  ji'iintx   ilnlihlif   rci-k'iHt'tL 

92.  Poles  and  polars.  The  /"•»/<//•  y//////<-  of  a  point  //,  (the  /<«/,  j 
with  respect  to  a  surface  of  the  second  order  whose  equation  is 
(  1  ).  vj  I"',  is  defined  as  the  plane  whose  coordinates  are 


The   following   theorems  are   obvious   or  mav   lie   proved  as  are 
the   >imilar  theorems   of       :>4: 


7.   If  tin-  i  a  ilf  ix  "a  //K'  nt/rf'1'1''.  f/f  jmliir  jiliii 
tin-  fi'J,-   li.-iii'i   tin    f,ntnt   if  .'unt.irt. 

II.      T"     l   >''/'>/    }l'i!llf     ll'it     ll     XI  /l</lllll  I'     l><iillf      'it'    ///,'      Kill-fill1!'      '•"/•/•<'.vy)-///r/.V 

ii  tini'jUi'  juiliir  jilii/n  . 

III.  T'i  i  >•!',->/  jil'Di,'  t't>n'i'x]ni)nl>*  'i  iini'iiii    ji"li    ii'Jn  ,i  ,i,i,l  'i)i///  ii'ln  n 

tin     1/1.1,  '/-i  niili'l  lit    "t    tji,     xli'i'tili-i     '/»•  >•    //'//    1'tinisli. 

IV.  .1    //"A//'    i  il<  Di  i-    I'uiililiiix    its    ji'Ji     irjn  ii    it//,/   <,nli/    >r]n  >i    tli,'    ji'il,-    is 
fill  tin'  >•?//•;',/,-,. 

V.  .!//   /•"/-//•    jthnii-x   jinx*    tln-oii'/li    nil    tin     m'n>/ii/<ir   [mint*    »/   //,,• 

tiUl'lili'i'     H'lnii     ts  II  I'll     i    list. 

VI.  If  'i  /'"'nit    /'  fi.'n  mi   tin    i>',l,ir  /Jinn'  'if  <i  i>"int   <t>.   tin  n   <t>  h'<-x 
mi  ///,•  jinlnr  1,1,1,1,    ,,f  I'. 

VII.  .!//  tnniji'iil  j'/.i/i,*  tfn-'ni'jfi  <i  /»>//)?   /'  /"//'•//   ///••  xurftfr  in   'i 

<•!!,•'•'       "'I,,','/,     //'is     in     tli,      y,  .,/.//-   y,  /,//,,       ,,/'     /'. 


STKKACKS  OK  SKCONI)  oKPKi;  AND  SKC()M>  CLASS     "2'2:\ 

VIII.  /•'"/•  <t  xi/rtiK'f  >if    xt<<-nn<i  "/•»/('/•   iritlimit   Ki'tii/u/ni'  /«//'///>•   it   /.„• 
pnxxihl?   iii   'in    in fin/ff   number  "J    triii/x  f<>   I'mixt  ni'-t   i>   ti'fr>t/n-ilr»n   >',/ 
irjifi'/i   1'in'h    fiii-i-   ix   f/n'  jmliir  jiluui'   <>f  f/n-   nppnxitf    /•»•/•/»'./-. 

These  arc  xelf-p<>lttr  t^tniJn'^rniix. 

IX.  If  n»ij  xtrnii/lit  1 1 in-  ni   >x  puxxt'il  t/ii'iiiti//!   'I  [mint    I'.   ii/nl   /,'  iiinl 

,S'      (//'('       ///,'       IllltllfX       III       H'/ltl'//       Ill       I  lltl'I'Xl'l'tx       ll       ftllftlll'l'       »t       Xl    l',l)lll      I, I'!/,'/'       1/11,1 

<)  in  f/n'  j>»i//t  at'  inti'rxi'i'ti'iiH  <>f  in  iiml  tin'  i>«liir  i>l<nn'  >>t'  I\  ///>/i    /' 
ii/nl  <t>  i/ri'  humi'iiii'1  I'onj injittt'x  irith   r^xju-i't  t,>   /,'  <///,/   ,s'. 

In    addition    to    these    theorems    \ve    will    state    and    prove    the 
following,    \vhieh   have   no  counterparts   in    ^  -\  [  : 

X.  Tin'  /n'/'/r  ['linn1*  nf  pntntx  on  <t  r<i/i</>' J<>nn  *i  jn'tn-i/  <>f  jiJmn'x  fin' 
,r,n'x  «f  //"///-•//  ix  i-itlli'il  f/n-  fi»tjtli//lt<'  />"/>//•  Jim1  <>/  tin'  fnix,'  <>t'  //,,•  r<iii<i>\ 
Hfi-lj >/-i>fiilIi/  tin'  ji'ilnr  i>!<DK'x  tif  jmitltx  "ii   tin1  n.rix    <>('  tliix  jn'iifil   /'«/•/// 
ilnntlir/-  Jn'll/'ll    f/n'    il.rix   at    K'Jnt'fl    ix    f/n'    /ntxi'    nt    I  In1    ni'l'/l  iliil    /'ii/n/i'. 

Consider  anv  rani^e  two  of  whose  points  arc  /'  and  n  (!•"!'_;•.  ")•")). 
Let  the  polar  planes  of  /'  and  <t>  intersect  in  I.K.  and  let  ./  he  anv 
point  of  LK.  The  polar  plane  of  .1  must  contain  hoth  /'  and  (,> 
(theorem  V I  )  and  hence  the  entire  line  l'<t>.  Now  let  I!  he  anv 
point  on  r<t>.  Its  polar  plane,  must 
contain  .1  (theorem  VI).  Hut  A  is  ,^ 

anv  point  of  l.I\.  Therefore  the  polar 
plane  of  //  contains  /,  l\.  This  proves 
the  theorem.  It  is  to  be  noted  that  the  / 

opposite  ed^es  of  a  self-polar  tetra- 
hedron are  conjugate  polar  lines.  /,/__ 

XI.  If  fll'n  cniy  lit/lift'    flii/iir   /llli'X    III- 

ti  /-xi-i-f,  1'iii-h  ix  f'Dir/i'/it  fn  fin*  x/irt'i/i-i' 
itt  t/n'/'r  tin'mt  nf  i/ift'/'xi'/'f >n)i. 

Let   t\\'o  conjugate  jiolar  lines,  /'n 
and     U\.    intersect     at     It.     Since     // 

lies  in  each  of  the  lines  l'<t>  and  I.l\  its  polar  plane  must  contain 
each  of  these  lines  hv  the  definition  of  eon  jugate  polar  lines.  1 1  cine 
the  polar  plane  o|  /,'  eoiitains  //  and  is  therefore  (theorems  1\ 
and  I  )  the  tangent  plane  at  //.  The  t  \\  o  lines  I.l\  and  /'','  1\  inu 
in  the  tangent  plane  and  passing  through  //  arc  tangent  to  tin- 
surface  at  /«'. 


±J4  IT  I  K  KK  - 1  >I  M  KN  Sl<  >N  A  L   ( !  KOM  KTK  V 

EXERCISES 

1.  Show  that  anv  chord  drawn  through  a  fixed  point  /',  intersecting 
at  infinity  the  polar  plane  of  /'  with  respect  to  a  quadric,  is  bisected  by  /'. 
Hence  show  that  if  a  qiiadric  is  not  tangent  to  the  plane  at  infinity  there 
is  a  point  such  that  all  chords  through  it  are  bisected  by  it.  This  is 
the  i'i  nf>  i-  of  t  he  quadrie. 

'2.  Show  that  the  locus  of  the  middle  points  of  a  system  of  parallel 
chords  is  a  plane  which  is  the  polar  plane  of  the  point  in  which  the 
parallel  chords  meet  the  plane  at  infinity.  This  is  a  <li<i nn'trul  /i/</nr 
conjugate  to  the  direction  of  the  parallel  chords.  Show  that  a  diametral 
plane  passes  through  the  center  of  the  quadrie,  if  there  is  one,  and 
through  the  point  of  contact  with  the  plane  at  infinity  if  the  surface 
is  tangent  to  the  plane  at  infinity. 

3.  Trove  that  all   points  on  a  straight  line  which  passes  through  the 
vertex  of  a  cone  have  the  same  polar  plane  ;   namely,  the  diametral  plane 
conjugate  to  the  direction  of  the  line. 

4.  Show  that  if  a  plane  conjugate  to  a  ^iven  direction  is  parallel  to 
a  second   Ljiven  line,  the  plane  conjugate  to  th"  latter  line  is  parallel  to 
the   first.     Three   diametral    planes  are   said   to   be   I'niijiujnti'  when   each 
is  conjugate  to  the  intersection  of  the  other  two.     Show  that   the  inter- 
sections of  three  conjugate  diametral   planes  with  the   plane  at   infinity 
form  a   triangle   which   is  self  polar  with   respect   to  the  curve  of  inter- 
section of  the  qiiadrie  and  the  plane  at   infinity.     Hiscuss  the  existence 
and  number  of   such  conjugate  planes  in  the  two  cases  of   central  quad- 
rics  and  quadrics  tangent  to  the  plane  at   infinity. 

5.  Show  that    if  a   line   is  tangent   to  a  qnadl'ie  surface   its  conjugate 
polar  is  also  tangent   to  the  surface  at   the  same  point,  and  that   the  two 
con  jugate  polars  are  harmonic  con  pirates  with  respect  to  t  he  1  wo  lines  in 
which  the  tangent   plane  at  their  point  of  intersection  cuts  the  surface. 

G.  Slmw  that  the  conjugate  polars  of  all  lines  in  a  pencil  form  a 
pencil.  When  do  the  two  pencils  coincide'.'  Show  that  the  conjugate 
polars  of  all  lines  in  a  plane  form  a  bundle  of  lines,  and  conversely. 

93.  Classification  of  surfaces  of  second  order.    With  the  aid  of  the 

results  of  the  last  two  sections  it  is  now  possible  to  obtain  the 
simplest  equations  of  the  various  tvjies  of  surfaces  of  the  second 
order  which  have  already  been  arranged  m  classes  in  ^I'l. 

I.  Tin'  i/i'ii,  i-iil  nu/'t'iti'i;  A  •'  <>.  The  si  i  rface  1  ias  1 1  o  si  1 1  o'u  1  ar  ]  mi  11 1 
(  ^  '.'1  »  and  there  can  be  found  ^elf-polar  tetrahedrons  (  sj  '•'-).  Let 
one  such  tetrahedron  be  taken  as  the  tetrahedron  of  reference  in  the 


SI'  KI-'ACKS  OF  SKrnND  <M;i>KK  AND  SKniNP  CLASS     -J-J.", 

coordinate  system.  '1  lini  the  equation  tit  tin'  surface  must  he  such 
that  the  polar  of  0:0:0:1  is  st=  0,  that  of  U  :  (j  :  1  :  0  is  ./  <>. 
that  of  0:1:0:  0  is  ./•.,=  <>,  ;Ui,l  that  «,f  1:0:0:  <>  is  ./  1  =  <>.  The 
equation  is  then 

"n-'T  +  ":;•'••;  +  ".,«•'•.;  +"«-'Y  •  "•  (-; 

where  no  one  of  the  coetlicients  can  lie  /cn>,  for,  if  il  \vriv.  the 
surface  would  contain  a  singular  jmint. 

Il  is  obvious  that  if  the  original  tetrahedron  of  reference  were 
real  and  if  the  coelVicients  in  the  original  eijiiation  of  the  surface 
were  real,  the  new  tetrahedron  of  reference  and  the  new  coefficients 
are  also  real.  \Ve  mav  now  replace  ./•,  in  the  last  equation  l>v  <i  u  .i\ 
and  have  three  types  according  to  the  si-^ns  of  the  terms  resulting. 

1.   Th>'  ////<'///'/"//•//  ////»',      ./•;'  4  ./•.:  4  •''.["'  4  •'',"  ---  "•  (•>) 

This  equation   is  satisfied  hv  no  real  points. 

•2.  '/'/»'  >»-»l  t  '/}>,;  ./v  +  .'7  +  .''f  -  .'V  =  (  '•  (  •*  ) 

No  real  straight  line  can  meet  this  surface  in  more  than  two  real 
points.  If  it  did,  it  would  lie  entirely  on  the  surface  (  ^  Ml  ).  and 
hence  the  point  in  which  it  met  the  plane  ./•  =(>  would  lie  a  real 
point  of  the  surface.  Hut  the  plane.  /-^  0  meets  the  surface  in  the 
curve  ./••+.'.;'  -f  .'';f  =  ".  which  has  no  real  point.  Hence,  as  \\  as  said. 
no  real  straight  line  can  meet  the  surface  in  more  than  two  real 
points.  The  surface,  however,  contains  imaginary  straight  lines  as 
will  he  seen  later. 


Through  e\  erv  point  of  this  surface  -^o  t  \\  1  1  real  straight  lines 
which  lie  entirely  on  the  surface.  This  follows  from  the  fac_t  that 
whatever  lie  the  values  of  \  and  ft,  the  two  lines 


.''.,   -  .'•  -f-  /u  (  .r  +  ./•  )      o 

lie  entirely  on  the  surface.  Moreover,  values  of  X  and  /u  mav  he 
easily  ioiind  so  that  one  ot  each  ot  these  straight  lines  ma\  pass 
through  any  point  ot  the  surtace.  '1  Ins  matter  \\  ill  he  discussed  in 
detail  in  £  '.Hi. 

As  the   three   tvpes   of  sitrliiccs  here   named   are   distinguished    hv 
properties  which  are  essentially  different   in  the  domain   ot   ivahtv. 


22()  T1IKKK    IMMKNSlnXAI.   (iKOMKTRY 

the  corresponding  equations  can  evidently  not  be  reduced  to  each 
other  bv  anv  real  change  of  coordinates.  However,  if  no  distinction 
is  made  between  reals  and  imaginaries,  all  surfaces  of  the  three 
tvpes  mav  be  represented  by  the  single  equation 


II.  Thf  '•"//. x.    A  ~  0.  hut   not  all  the  lirst  minors  are  zero.   The 

surface  has  one  singular  point  ( sj  1*1  )  and  is  a  cone  with  the  singular 
point  as  the  vertex.  Let  the  vertex  he  taken  as  A  (0:0:0:1). 
Then  in  the  equation  of  the  surface  11=11  —a.  = /f  =  0.  Take 
now  as  #(0:0:1:0)  anv  point  not  on  the  surface.  Its  polar  plane 
contains  .1  (theorem  V.  £  t<'2  )  hut  not  />'(  theorem  IV,  ^  '.»-)•  Take 
as  r (0:1: 0:0)  anv  point  in  this  plane  hut  not  on  the  surface. 
Such  points  exist  unless  the  polar  plane  of  />'  lies  entirelv  on  the 
surface,  which  is  impossible  since  /•'  was  taken  as  not  on  the  surface. 
The  polar  plane  of  ('  contains  A  and  /.'  and  intersects  the  polar 
plane  of  />'  in  a  line  through  A.  Take  I>  (  1  :  0  :  0  :  (I  )  as  any  point 
mi  this  line.  We  have  now  fixed  the  tetrahedron  of  reference  so 
that  0:0:0:1  is  a  singular  point,  the  polar  plane  of  0  :  0  :  1  ;  (.1  is 
./•  =  (I,  the  polar  plane  of  0:1:0:0  is  ro=  0,  and  the  polar  plane 
of  1  :  'I  :  ii;  n  js  ./•  0.  Therefore  the  equation  of  the  surface  is 

"n-'Y-f  ",.Xr -f  ":M •*•;?=  °- 

where  no  one  of  the  three  coefficients  can  vanish,  since  the  surface 
has  onlv  one  singular  point.  P>v  a  real  transformation  of  coordinates 
this  equation  reduces  to  two  tvpes: 

1.    Tin'  innii/iiKirii  <•»/>>'.      ./',"  +  ->'^  +  •''.•"' =  "• 

•2.  Tin-  fi-ttt  <-nn>\  .>•{+  ./-.:-  r.f=  0. 

III.  T'/'»  >i/f>Txi<'f//i</  jilinn-x.    A ~  0,  all  the  first   minors  are  /ero, 
hut    not    all   the  second   minors  are  /ero.     This  has  heen  sulh'cientlv 
discussed    (^'.*1  ).     There   are    ohviouslv    two   tvpes    in    the   domain 
of    reals  ;    naiiielv  : 

1.    I iiniiji /i'i n/  jilii/nx,  ./^--f- ./•'-'—  0. 

•2.  /•'..//  i>/<mrx,  ./-,-  -  ./•:-    o. 

I\'.    '/'//•-/  t'liiiK-iih-nt  /'/<i/iis.     A       0.  all  the  lirst   and   all  the  second 
minors   are   equal    to   y.rro.     Kvidently   the    eijiiation    in    this  case   is 

re(|iieihle  to  the   funu 

'" 


Sl'RKACKS  (>F  SFJ'oM)  oKDKK  AND  SKCoND  CLASS     '1'1~ 

but  the  plane  ./'  =0  is  lint  necessarily  real.  In  tact  the  condition 
tliut  all  the  second  niinors  ol  A  vunisli  is  tin-  condition  that  the 
left-hand  member  of  equation  (  1  ),  vj  '.MI,  should  be  a  perfect  >quare, 
as  is  easily  verified  bv  the  student. 

94.  Surfaces  of  second  order  in  Cartesian  coordinates.  As  \\  e 
have  seen  (  vj  *'2  ),  we  obtain  ('artesian  coordinates  l'r<>m  general 
<|iiadri[)lanar  coordinates  liv  taking,'  mie  oi  the  coordinate  planes  as 
the  plane  at  infinity  and  i_n\in^  special  vahie>  to  the  eiin>taiits  /•  . 
'1  his  being'  done,  the  general  equation  <>|  the  second  decree  will 
be  written 

'i.r+f>.i--  -f  '•;•"+  'Ifir-  +  -//:•./  '+  -//.'•//  -f-  -/.//  +  -2  //////  i  -ln\t  -u.  ,//-.  o,  (]  , 
which  reduces  to  the  usual  noiihomoj^eneous  form  \\heii  /  i>  placed 
etjual  to  1. 

Fur  equation  (  1  )  the  results  of  JJJJ  '.Hi  <»:;  remain  unchanm-d 
except  for  a  slight  chaiiLTe  of  notation.  \Ve  \\ill  refer  tn  the  ctiua- 

L  *T>  i"  1 

lions  ot  these  sections  bv  number  and  make  the  neeessarv  change 
in  notation  without  further  remark.  A>sumin'_;'  that  A  ;  "  \\  e 
ma\'  find  the  pule  of  the  plane  at  intinitv,  I'm-  example,  hv  placing 
>it  in  eijtiations  (  1  ).  ^  I1'-',  equal  to  the  coordinate^  0  :  U  :  '.)  :  1  of  the 
plane  at  intinitv.  There  result  the  equations 

(I.C  +  ////    +  </•    -)-  It      .-~  <», 

//./•  +  l<i   +  t'-  +  nit  --  u, 

(2) 

//./•  +.///    +  '••    +  nt    =  U, 
/./•    +  in.1!  +  HZ  +  i/t    -  -  p. 

tlie  solution  of  which  is  the  coi'irilinates  of  the  pole  required.  Thi> 
pole  is  therefore  a  linite  point  when  the  determinant 


is  not  /.ero  and  is  a  point  at   in  tin  it  \    when    /'       <  '. 

In  the  latter  case,  \>\  theorems  I  V  and  1.  ^  '.''_'.  the  surface  i> 
tangent  to  the  plane  at  intinitv.  In  the  former  case,  it  the  pole 
of  the  plane  at  infinity  is  taken  as  H:0  :():!.  then  /  //  ''. 

and  consequently  it  appear.^  that  if  ./^  :  //1  ::,:/,  is  a  point  on  the 
surface,  —  j'  :  —  //_:  -  :.  :  /.  i>  aUo  on  the  surface.  'I'he  point  is 
therefore  called  the  <-,-„(,  r  of  tin-  Mii't'aec.  and  the  >urface  i-  called 


228  THREE-DIMENSIONAL  GEOMETRY 

;i  <->>nti'<il  #iirt'<ii;-.  Conversely,  it  a  surface  without  singular  points 
has  a  ('filter  (that  is,  if  there  exists  a  point  which  is  the  middle 
point  of  all  chords  through  it  ).  that  point  is  the  pole  of  the  plane 
at  inlinity.  This  follows  from  theorem  IX,  vj  I*!*,  or  may  be  shown 
bv  assuming  the  center  as  the  origin  of  coordinates  and  reversing 
the  argument  just  made. 

We  have  reached  the  following  result: 

A  .vi/rt',1,-,  <>t'  x,  <;,/i, 1  nfijt'i-  trit/i  (lie  I'lfuatinn  (1  )  i*  «  <'<'ittr<il  unrfitce 
"/•  <i  nmti-t -nti'itl  xttrtiirf  <ii-i;i/;/tit,/  <tx  (hf  tlt'tt't'/iiitiiliit  I>  in  /ft  <>r  ix 
,<initl  /<-  zct'ij.  A  itmicoiti'iil  xiirt<(<-<-  <*  tidcji'Ht  t<>  th<'  j>l<ini'  <it  infinity, 

Holding  now  to  the  significance  of  the  determinant  A  as  given 
in  vj  i)()  we  may  proceed  to  lind  the  simplest  forms  of  the  equa- 
tions of  the  surface  in  ('artesian  coordinates.  There  will  be  this 
ditYerence  from  the  work  of  ^  (.'o  that  now  the  plane  /  =  0  plays 
a  unique  role  and  must  always  remain  as  one  of  the  coordinate 
planes.  The  other  three  coordinate  planes,  however,  mav  Vie 
taken  at  pleasure,  and  we  shall  not  at  present  restrict  ourselves 
to  rectangular  coordinates. 

1.  ('i/tf/-'il  snrJ\i'-,'K  without  xiiti/uhir  jioints.  As  in  Jj  '.'•<,  bv  refer- 
ring the  surface  to  a  self-polar  tetrahedron  one  of  whose  faces  is 
the  plane  at  intinitv  its  equation  becomes 

".'•'+  t>>f+  <-r+  <1t-=  0. 

According  to  the  signs  of  the  coefficients  this  gives  the  following 
tvpes  in  nonhomogeneotis  form: 


,.,.      .          .  ,,.        .  ,  ./•"      //"      ~~          _, 

( 'i  )    1  he  imagmai'V  ellipsoid,  -f- '   ,+    .,=  — 1. 

(/-  )    The   real   ellipsoid.  +'.,+  "     =  1. 

(<•)  The  hvperboloid  of  two  sheets,  =1. 

•'•'      /       :'J=   ^ 

.  *  /  ' 

./"       //        i- 

II.  \"in'>  nt rill  }<iir1'iii-tx  ir'itfiKiit  xiniiiil'ir  1'ii'uitx.  Since  the  plane 
at  intinitv  can  no  longer  be  a  face  of  a  >elf-polar  tetrahedron,  we 
cannot  use  the  method  of  jj  \*'.\.  We  \\  ill  take  the  point  of  taiigeiicv 


SUMACHS  OK  SE('oXI)  ORDKK  AND  SKCoND  CLASS     1^1 

in  the  plane  at  infinity  as  /.'  (  0  :  o  :  1  :  o  ).  Then  //=./'  c  "  ami 
n  'r-  0.  Take  an  arbitrary  line  through  />'•  It  meets  the  surface 
in  one  other  point  .1,  which  \\  e  take  as  0:11:0:1.  \\Y  then  take 
the  tangent  plane  at  .1  as  2  =  0.  Then  /  =  m  =  d  =  0,  and  the 
equation  of  the  surface  is 

</./•- -f  _  //./•//  -f  /•//-  +  <•/-=  0. 

The  tangent  plane  at  .1  meets  the  plane  at  infinity  in  a  line 
(,?=(),  /  —  0),  \vhieh  is  the  conjugate  polar  to  the  line  .I/.'  (./•--  (), 
//  —  0  ).  The  points  ('(0:1:0:0)  and  l>  (1:0:0:0)  mav  he  taken 
as  anv  t\\'o  points  on  this  line  such  that  eaeh  lies  in  the  polar 
plane  of  the  other.  Then  h  —  0,  and  the  eijuatioii  of  the  surface  is 

reduced  to 

</./"  +  /-//--f  //•'  —  0. 

According  to  the  signs  which  occur  we  lui\  e  two  t\'pes: 

1.  Th.'  oval  type: 

The  elliptic  paraboloid,  ,-f  '     =  nz. 

<r      f>~ 

2.  The  x.t.lll,'  ti/jH-: 

The  hvperljolic  paraboloid,      '  .,—  '   ,=  nz. 

<r      l~ 

The  discussion  of  surfaces  with  singular  points  presents  no  features 
essentially  different  in  ('artesian  coordinates  from  those  found  in 
the  general  case.  If  the  surface  has  one  singular  point,  it  is  a  cone 
if  the  singular  point  is  not  at  intinitv  and  is  a  cylinder  if  the  sin- 
gular point  is  at  inlinity.  If  the  surface  has  a  line  of  singular 
points,  it  consists  of  two  intersecting  or  two  parallel  planes  accord- 
ing as  the  singular  line  lies  in  finite  spare  or  at  inlinity.  If  the 
surface  has  a  plane  of  singular  points,  it  consists  of  a  plane  doubly 
counted,  which  may  he  the  plane  at  inlinity. 

95.  Surfaces  of  second  order  referred  to  rectangular  axes.  In  the 
preyiotis  section  no  hypotheses  were  made  as  to  the  angles  at  which 
the  coordinate  planes  intersected.  For  that  reason  the  coordinate 
planes  leading  to  the  simple  forms  of  the  equations  could  be  chosen 
in  an  infinite  number  of  ways.  \\Y  shall  now  ask  whether,  among 
these  planes,  there  exist  a  set  in  which  the  planes  ./•  o.  //  0, 
and  r  -0  are  mutually  orthogonal. 


230  TIHlKK-niMKNSloNAL  (JEO.MKTUY 

Consider  first  the  central  surfaces  without  singular  points  for 
which  />  -  0.  Tin-  plain1  at  infinity  cuts  this  surface  in  the  geu- 

„,.-  +  L,r  +  ,-z-  +  -2/1/2  +  -2 uzjT  +  -2  /,.,-,/  =  0,  (1 ) 

\vherc  ./• :  >/ :  z  are  homogeneous  coordinates  on  the  plane  /  =  0. 

When  the  equation  of  the  surface  is  referred  to  a  self-polar  tetra- 
hedron of  which  the  plane  at  infinity  is  one  face,  the  eiirve  (1)  is 
referred  to  a  self-polar  triangle.  If  the  axes  in  space  are  orthogonal, 
the  triangle  must  also  he  a  self-polar  triangle  (theorem  V,  vj  Si) 
to  the  circle  at  infinity 

•r+y'+r^o.  (-2) 

( )nr  problem,  therefore,  is  to  find  on  the  plane  at  infinity  a  triangle 
which  is  self  polar  at  the  same  time  with  respect  to  (1  )  and  (  2 ). 

P>y  £  4o  this  can  he  done  when  and  only  when  the  curyes  (1) 
and  (_)  intersect  in  four  distinct  points  or  are  tangent  in  two 
distinct  points  or  are  coincident. 

In  the  first  case  there  exists  one  and  only  one  self-polar  triangle 
common  to  (1  )  and  (  '2  ),  and  therefore  there  exists  only  one  set  of 
mutually  orthogonal  planes  passing  through  the  center  of  the  quad- 
ric  and  such  that  by  use  of  them  as  coordinate  planes  the  equa- 
tion of  the  quadric  becomes 

„•./•- -f- 1,,1 4-  ,-r  +  ,7  =  0.          <  «  =t=  I  -  <•  =t  0 ) 

These  planes  are  the  principal  diametral  jJiun-x  of  the  quadric, 
and  their  intersections  are  the  principal  n.ri:x. 

In  the  second  case  there  are  an  infinite  number  of  planes  through 
the  origin,  such  that  by  use  of  them  as  coordinate  planes  the  equa- 
tion of  the  (juadric  becomes 

a  (  .r  +  //'J )  +  >•--  +  -/  =  0.  (  ,/  --*=  ,-  ~   0  ) 

Here  the  axis  OX  is  fixed,  but  the  axes  O.V  and  O  Y  are  so  far 
indeterminate  that  they  may  be  any  two  lines  perpendicular  to  OX 
and  to  each  other.  The  surface  is  a  surface  formed  by  rcyolying 
the  conic  'i.r+  '•••-+  '/  ---  0.  //  ---  0  about  <>/. 

In  1  lie  t hi i'd  case  any  set  of  mutually  perpendicular  planes  through 
the  origin,  if  taken  as  coordinate  planes,  reduce  the  equation  ot  the 

iiuadric  to  the  f< n-m 

-""       * 


sniFACES  OF  SKCoND  <>K1>FK  AND  SECOND  CLASS     '2:\\ 

It  is  t<>  !>»•  noticed  that  if  tin-  coefficients  iii  equation  (  '2 )  iin- 
n-uK  one  ot  tli''  iibovu  cases  lU'ci'ssiirilv  occurs.  !•  or  in  this  case 
the  solutions  of  equations  (1  )  and  (  '2  )  consist  of  inmgiiwrv  points 
which  occur  in  pairs  as  complex  imaginary  points. 

If  \ve  consider  the  iioiicentral  qiuulpies  without  singular  points 
and  use  the  notation  of  J;  '.' 1,  \\  e  notice  tirst  that  if  the  axes  of 
coordinates  are  rectangular,  the  point  //cannot  IK-  on  the  circle  at 
inlinitv,  since  the  line  ('/>  must  l»e  the  polar  of  ]i  with  respect  to 
the  circle  at  inlinitv.  I  he  point  I'>  bein^'  lixi'tl  liv  the  ijuaih'ie  sur- 
face, the  line  C/MS  then  fixed,  and  consequently  the  line  A  l>,  since 
All  is  the  conjugate  polar  of  Cl>  with  roped  to  the  qiiadric.  The 
point  .1  is  then  fixed  and  is  called  the  /v/vV.r  of  the  (piadric. 

The  points  ('  and  1>  must  MOW  be  taken  as  conjugate,  both  with 
respect  to  the  circle  at  inlinitx  and  with  respect  to  the  conic  ot  inter- 
section of  the  qiiadric  and  the  plane  at  inlinitv.  If  the  two  straight 
lines  into  which  this  latter  conic  degenerates  (  cf.  Kx.  1,  ^  sti  j  are 
neither  of  them  tangent  to  the  circle  at  intinitv,  the  points  ('  and 
/>  are  uniqiielv  fixed.  If  both  of  these  lines  are  tangent  to  the  cir- 
cle at  infinity,  the  point  ('  may  be  taken  at  pleasure  on  ('/>,  and  l> 
is  then  fixed. 

In  the  first  case  there  is  one  tangent  plane  and  two  other  planes 
perpendicular  to  it  and  to  each  other,  by  the  use  of  which  the  equa- 
tion ot  the  ijiiadric  is  reduced  to  the  lorni 

(/./•- 4-  /,</--    us.  (n   -    /• ) 


In  the  second  case  then-  are  an  infinite  number  ot  mutuallv 
oilho^onal  planes,  one  of  which  is  a  fixed  tangent  plane,  bv  the 

Use    o|'    \\hicll    the    eijllatioli    of    the    (ptudl'ie    is    reduced    to    the    form 

H  (  .>'"+  //"  )         >IZ, 

and   the  (jiliidrie   is  a   paraboloid   ot    pi-volution. 

In  all  other  cases,  namelv,  when  the  point  ot  taii'_;vncv  ot  the 
(piadric  \\iih  the  pi, me  at  mtiniu  is  on  the  circle  at  mlimtv  or 
\\heii  the  section  ot  the  i|iiadptc  \\ith  the  plane  at  intinitv  consists 
ot  two  straight  lines,  one  and  onl\  one  ot  which  is  tangent  to  il.e 
cii'i-le  at  infinity,  the  equation  oi  the  surtacc  <'annot  be  reduced  to 
the  above  forms  bv  the  use  of  rectangular  axes. 

It'    the    coefficients  of  the   lel'lllS  of  the  second    order  ill   the    equation 

of  the  iiuadric  arc  real,  the  rectuii'''ulai'  axes  alwa\s  exist. 


232  THREE-DIMENSIONAL  (lEOMETKY 

EXERCISES 

Examine  the  following  surfaces  for  the  existence  of  principal  axes: 

2.  L'  ./•-  +  (  1  +  /)//-+  ;;-  +  (  1  +  /)./•//  =  0. 

3.  .r-  +  2/X-+  7  .-.-+   I  ///::  +  1  =  0. 

4 .  L'  .!•-  f  -.-  +  1'  /./•//  +  1  =  0. 

5.  ;i  ./•-  +  L'  //-  +  7  :.-  +  0  ///.v  +1-0. 
G.  ./•-+  2 /.r//  -  //--  ,v-+  1> :;  =  0. 

7.  ./•;.  +  ///.-:  +r  =  0. 

8.  ./•-  -  L'  /./•//  +  //-+  1'  ./•  +  L'  ,v  =  0. 

9.  Examine  the  quadries   with  singular  points  \\\   the   methods  of 
this  section. 

96.  Rulings  on  surfaces  of  second  order.  \Ve  have  seen  (£  \\'-\) 
that  the  equation  of  any  surface  of  the  second  order  without 
singular  points  can  he  written  as 

it   no  distinction  is  made  hetween  reals  and  iniaginarics  or  hetweeii 
the   plane  at   iiilinity   and  any  other  plane.     This  equation   can   he 


X  ,.,, 

A.,  (_) 

.,, :  _**=„  (3) 

./•  —  /./•  ./•  —  t.l' 

:i  -t  1  2 

whence  follows  for  anv  point  on  the  surface 

./ 1  :  ./-., :  ./  , :  ./'4  -----  X/z  +  1  :  /  ( '  —  X/x  +  1  )  :  X  -  fj.  :  /  ( X  +  ^.        (  4) 
l-'roin  these  (Mpiatioiis  the  following  theorems  are  easily  proyed  : 

/.  <>ii  <t  xii i't'ii'-i'  «f  si'foml  "/•'/-•/•  tt'it/tmit  xi>ir/n/<ir  /^////.v  //,-  t//'<> 
t'ltntiliix  nt'  xtrii'n/til  ////ex,  nn,'  i/i'fitii'i/  In/  I'l/udtt'iitx  (-)  diiil  f/n'  <>(/n'r 
In/  i  ninttinnx  (  '•'>  ). 

For  if  X  is  '4'ivcii  any  constant  value  in  (  "J  »  the  eipiations 
represent  a  slrai<_dit  line  e\ci'\'  jioint  ot  which  satislies  cijiiation  (  1  ). 
SimilarK  .  fj.  mav  lie  L;i\-eii  a  con>iant  \aliie  in  (-1).  The  straight 
lines  (-2)  and  (  '•'>  )  are  called  '/<//,/•/>/<//•*. 


SCKKACKS  OK  SKCOND  <»KDKR  AND  SKCoND  CLASS     '2:\:\ 

II.  Tlll'iilHjh  i'ili'/l  Jin/lit  "J~  tin  Xlirt'iti'i'  i/ni  X  "in'  ill/-/  null/  "in  Hi" 
nt'  ,  ili'/l  f'l nil 'I '_//. 

For  an\'  point  r  of  the  surface  determines  X  and  /z  uni(|Uely. 
///.  l-'iifli  lint'  "t'  "II,'  fit/nil  i/  i  iit>-rx,'''t  x  ,  ilfli  //it,  "f  tin  "tin  r  t'liinil  i/. 
For  anv  pair  of  values  of  X  and  p.  leads  to  the  solution  (  I  ). 

IV.  X"   tn'"   ///n  x  "f  th,'   xiim,'    t'ttinilii   iitti  rxi'i-t. 
This   is  a  ci  irollar\'  tot  heoreiu  1 1 . 

V.  A    f't/ii/i'tif   /i/'i/ti     'it    'In  if   /'"'nit    "t'  tin     xiirt'tln'    int,  rx,  ,-tx    tin     xtn'- 
t'tifi'    in    tin1    tii'n   1/,-in  i'iit"i'x    tln'"ii<jh    tlntt   jiniiit. 

For  the  two  generators  arc  tangents  and  hence  he  in  the  tangent 
plane,  lint  the  intersection  of  the  tangent  plane  with  the  surface 
is  a  curve  of  second  order  unlos  the  plane  lies  entirely  on  the 
surface,  which  is  impossible  since  the  surface  has  no  singular  points, 
llciice  the  section  consists  of  the  two  ovnerators. 

VI.  Til,'  xil  rfiii','   f"iitiliiix  it"   "tin  r  xt /-iiii/lit   li m 'x  lit, 111   tin'   ,/,  n,  /'<//«/•,<. 

For  it'  there  were  another  line  the  tangent  plane  at  am  point 
of  the  line  would  contain  it,  which  is  impossible  bv  theorem  V. 

VII.  A  ni/   /i/it in'    tJi/-"iii/li    <i    i/,ni-/'ii/",'    intii-xii-tx    tin    xi/rt'ii'-i     i//x"    in 
if    </>  if  /•'//"/•    "t'  tin     "tin  r    t'tlinUi/    inn/    lx    ti/nif,  nt    t"    tin     xn rt'ti <•>'    'it    tin 

imtnt    "I    i  nt  i  /'xii't  mn    "t   tin'   tti'"   i/i'/ii' /'iif"/'x. 

('onsider  a  plane  through  a  generator//.  Its  intci sect  ion  with 
the  >urfaec  is  a  curve  of  second  order  of  which  one  part  is  kno\\u 
to  be  i/.  The  remaining  part  must  also  be  a  straight  hue  //.  \\hich 
is  a  n'eiierator  b\  theorem  \'l.  Since  //  and  //  are  in  the  same  plane 
t  hev  intersect  and  hence  belong  to  different  families  bv  t  heon-m  1  \  . 
The  lambent  plane  at  the  intersection  of  //  and  </  contains  these 
hues  by  theorem  \  and  hence  coincides  \\ith  the  original  plane. 

VIII.  It'    1  it'n     jitiii-ilx    til'   [limit'*     H'itll     tlnir     il./ix     ,/,  in  ,;/t",-x     ,,f    tin' 
xiiuif    til///////    i//-i'    ln'"(i'llit     int"    il    "iii'-t"-"!!,'   '•"/•i',xji"iitl,in-,     X"   t/nif    t /i'<> 
i'"i-r,xji"inl  i  ml    nlilinx    //ifr/'Xi'i'f    in    <l    i/i/n  /'ilt"/'    "I    (In     "t  In  r    t'imtl'1,    tin' 

/•<  I  ill  l"ll      IX      I't'"  ll'i't  I  I'l  . 

Let  the  axes  of  the  l  wo  pencils  be  taken  as  ./•  o,  ./  "  and 
./•  --  ".  ./•  ;__  I)  respect  i\fl\.  Since  these  lines  lie  on  the  surface,  the 
e  1 1  u  a  1 1  o  1 1  (if  the  surface  has  the  form 

f  .1  ./     f    <•  ./'  ./'     f    i1  ./'  ./'-(-'•/./         * '. 


•2:\4  THKKE-DIMENS10NAL  UEUMKTKV 

The  equations  of  plam-s  of  the  first  pencil  are 

.'',  4-  X./'n  =  0 
and   those  of  the  second  are 

'a +  /"•<=  ()' 

If  two  such  planes  intersect   on   the  surface,   we   have 


\\  hich  proves  the  theorem. 

IX.  Tin1    tuti  /'Kt  I'ttonx   nt    tin'   rni't't'tlJtonJltli/  j>l<(llt't*   nt    1  U'o  jii'ii  it'i't  l  ft' 
[n  it'-tlx  nt  ithint'x  ti'lth  nnitt titfi'ni'i't tiiij  d.ci'x  </<-/<r/'i(/i'  <i  utii'titff  nt  tst'i'uiid 

"/•</-/•  ichi'-h    i-ontiitnx   tin'   fi/'n   <l.r<  x  «J   tin1  pt'iu'tlx. 

Let    the   two    pencils    lie    ./•  -)-  \.r   =  ()    and   .V   -f  a.r  =  0,  \\  here    the 

A  _  -i  4 

,        •  Up,   +  fi 

prdjeetive  relation  is  expressed  Lty  \  — 

Then  if  a  point  is  eonimon  to  two  corresponding  planes,  it 
satisfies  the  equation 

which  is  iilso  satisfied   bv  the  axes  of  the  pencils. 

X.  (Dualistic    to  VIII.}     Lin*-*    <>f  <>m-  j'nntilii   <>f  </!•//> nit< //-.-«•   <-nt   out 
1'i'oj,  ,-ft'r,-   rtiitijfx  on   mill  t ii'n  lini'X  of  tin'  of/i,/'  t'llinili/. 

As  in  the  proof  of  theorem  YIII,  let  ./•  --  U,  .r ,  —  0  l>e  a  generator 
of  the  surface  and  let  .*'.  =  0,  ./'  =0  lie  another  generator  of  the 
same  familv.  The  equation  of  the  surface  is  then 

'•  ./'  ./'    -f-  «'  ./'  ./'    -\-  i'  ./'  ./'    -j-  r  ./'  ./'    -   -   '  I, 

1      1       i     '          J      1      l     '          :;•_•:;'  I      •_'     4 

and  the  generators  of  the  second  familv  are 


A    '.Ti-neriitoi-  of   this  familv   meets  ./•       ",  ./•  _     n   in   the  point  where 

./   :  ./•        '•   -f  '•  'V  :       '•        '•  X  and  meets  ./        o.  ./•       (I  where  .r  :  .r  —  \  :  1 . 
ii'-.'.;i  i  i 

I  he  relation  i>  evidentlv  pro|eetive. 

XI.    (DualistiC   tO  X.)       Tin      ////»•.•«•     //7//'7/     <-»ll/H-rt    <;,,-,•>  */»///,///,_,/    /mints 
»_t     tii'.,    i>i'«j> -i-li  i'<      i'il inj,  K     irittt     nmiuiti  t'xi  i-tlinj    //«/M  .v    //-     mi     •/     nil I'filfi' 


Sl'KKACKS  OF  SKCOND  OKDKI!  AND  SKCuND  CLASS     ^:J.", 

Let  one  ranin-  In1  taken  on  ./•  =  <>,  ./•  =  (I  anil  the  other  on  ./•  =  (I, 
./•  —  <).  '1  hen  tlie  points  o{  the  t\vo  ranges  are  i^iven  on  each  base 

hv   the    equations   ./'..  -f  X./'4  --  0    and    J-+fJ.J'n—l).      Let     the    projective 
11  'J/x  4-/^ 

relation  be  expressed  by  A 

7ft  -f  6 

From  these  it  is  easy  to  eoinjuHe  thai  the  coordinates  ot  anv  point 
on  the  line  connecting  two  corresponding  points  ot'  the  two  ranges 
satisfy  the  filiation 


EXERCISES 

1.  I  M^tinLruish  hetween  the  eases  iii  \vhirh  the  ^ciieratiirs  are  ival  or 
iinau'iiiarv,  assuming  that  the  eijiiation  uf  the  i|Hadrie  i>  real. 

2.  \\lia1  are  the  ^eiiei'ators  of  a  >pliere  '.' 

3.  Distinguish   liet  \veen  a  central  ipiadric  and  a    nonceiitral  one   l.\ 
>linwiii^  that  fur  the  latter  type   the  general  urs  arc  parallel   t<>  a   plain- 
and  for  the  former  they  are  not. 

97.  Surfaces  of  second  class,    ('nnsider  the  equation 

V.I,,  :ii,>'k=  <>,          ('!,,-=  Ait)  (1  ) 

in  plane  coordinates.  This  is  a  special  case  ot  the  equation  dis- 
cussed in  x  ss.  liquations  ('•'>).  $  **.  which  determine  the  limit 
points,  become 

p.'\       .I.,//,  4-J  ,,//.,+  -  1,,,";,+  -'>4"r  (  /--=  1.   -.   ;5.   4  )  <  -  ) 

and  eq  H  at  ions  (  ~>  ).  >  ss.  \\'hieh  del  i  ne  the  singular  plan  ex.  I  KM  •nine 
.!,  ,//,  -f.  /..,?/.,  4-  .1,  //,+  J,.,/^  =  <».  (/  =1,  "2.  :'..  4  )         ('.}) 

I  !'  we  no\\    place 


we  have  to  distinguish  four  cases. 

I.  A  •"  n.    Htjuat  ioiis(  ^  )  have  then  a  single  solution  for  //,:»  :  >/  :  ",. 

which,  it'  substituted  in  (  1  >.  Lfives  the  equation  of  the  snrlace  en- 
veloped hv  the  extent  of  planes.  'I'his  equation  may  l>e  more  eon- 
N'enieiitlv  obtained  hv  replacing  (  1  >  liv  the  equation 


2o6  THREE-DIMENSIONAL  (JKOMKTKV 

obtained  from  (1)  by  the  help  of  (-).    The  elimination  of  ?/.  then 


inves 


I, 


which  is  the  equation  of  a  surface  of  second  order. 

1'nder  the  hypothesis  A—  0  equations  (•})  have  no  solution,  so 
that  in  this  case  no  singular  plane  exists.  It  is  not  difliciilt  to 
show  that  the  discriminant  of  equation  (4)  does  not  vanish. 

We  have,  accordingly,  the  following  result:  .1  />/atn'  fj-d'ut  <>f 
second  das*  icith  nonranisJiiny  discriminant  consists  of  planes  cnrd'>j>- 
in<i  a  surface  of  second  order  without  ssitujular  points. 

This  theorem  mav  be  otherwise  expressed  as  follows:  .1  snrf<t<'<' 
of  st'fom!  dass  without  sini/nfar  plants  is  also  a  s>irf<i<-i'  <>f  s^<-<>n<l  onlfr 
without  shii/ufar  pohifs. 

II.  A  =  0,  but  not  all  the  first  minors  are  x.ero.  Equations  ( :>)  now 
have  one  and  only  one  solution,  so  that  the  extent  (1  )  has  one  and 
only  one  singular  plane.  Let  it  be  taken  as  the  plane  (1:0:0:1. 
Then  A^t  =  A  =An4-=A  =Q,  and  equation  (1)  takes  the  form 

.•l,1M12-M,2?/ss  +  -V/a2+  2,l1,?/,?/,+  2v/1:,M,»/3+  2J23M2?/3=  0,    ((i) 
\\'here  the  determinant 


Iocs    not    vanish    owing    to    the    hypothesis    that    not    all    the    first 

rs   of   the   discriminant   (4)  vanish. 
The    elimination    of    ut    from    eijuations    d)    and    equation    ('>) 
iyes,    then, 


which   are   the    (-(illations   of   a   nondegenerate    conic   in    the    plane 


Sl'KFACFS  <>F  SKCONI)  <MM>KK  AM)  SKCoNI)  <'LASS     '2'M 

We  have,  accordingly,  the  result  :  .1  /</<///»•  ,.rt<nt  nf  WUH,I  ,•/,>.«,.>, 
it'ith  <>/t<  xin</nl<ir  plnnt1  t'linxftitx  ntf  iilnni'K  n't/'u'h  <tr>-  tiitn/i/if  /••  •/  ii"ti- 
i/i'i/i'/ii'/-nfi'  I'nnii'  ////////  in  (hi-  xi/ii/n/'ir  j>/<(tti\ 

The  t'(jUUti()ll  ot  the  plane  extent  mav  lie  considered  the  eqiia- 
tion  of  this  conic  iii  plane  coordinates. 

III.  A  --  0,  all  the  first  minors  are  /.em,  hut  not  all  the  second 
minors  are  /.em.  Equations  (  •! )  no\v  contain  only  t\vo  independent 
(•((nations  and  hence  the  extent  contains  a  pencil  of  singular  planes. 
It  this  pencil  is  taken  as  u  =  d,  nit—  ",  the  e(|uatioii  of  the  extent 

becomes 

Auuf+'2  •^,>ll",^-lj>'-  ()-  (7) 

where  the  determinant  .1  .(,,  -.I,',  docs  imi  \anish  liecanse  of  the 
hypothesis  that  not  all  tin-  second  minors  of  the  discriminant  (  1) 
vanish. 

Equation  (7)  factors  into  two  distinct  linear  factors  and  hence 
the  plane  extent  consists  of  two  bundles  of  planes.  The  elimina- 
tion of  nt  between  equations  (-)  and  (7)  j^ives 


0 

which  define   the  vertices  ot   the  two  bundles. 

\Ve  have,  ucconlillj^lv,  the  result:  .1  )il<tin'  i.rftt/t  i>f  a<'i'n)i,1  ••l<inn 
irilli  <i  in  iif/l  "/  ftllli/n!i!>'  /'/<///'  x  r'///.sv.s'/.v  n1  1  ii'n  liii/nlli  x  »t  />/<///.  >•.  ///« 

sini/llhll'    [a   Ili'll     In   Ill'l     til,'     /I,'//'-//     <•<>/// ///">/      f"      f/li'     tll'n     I, Ilil, II,  'ft. 

IV.  A-  <•,  all  the  first  and  second  minors  are  /.em.  but  not  all 
the  third  minors  are  /.ero.  Filiations  (  :)  )  contain  oiih  one  inde- 
pendent equation  and  hence  the  plane  extent  contains  a  bundle  of 
singular  planes.  If  this  bundle  is  taken  as  //,  -  d,  the  equation  of 

the    extent    becomes 

.(,,".-'        0,  (*) 

where    .!      cannot    be   /em   because   of    the    livjxtt hesis    that    not    all 
third   minors  ;  if  (  1  )  are  /.ero. 

Hence  we   have   the   result:    .1    /</,///,    -.//////   r//'  xc/-,,//,/  ,-/,/.v.v      •//},  ,i 

I, II, nil,'    tit' silli/Hl'tr  fi/KIH-X   i-iiHXIttfx   nt'  t/Hlt    fmtl'/li'    (/"»/./_//    /•«•/,•"/">/. 

It  mav  be  noticed  that  the  elimination  ot  u.  betueeii  cijiiatioiis 
(ll)and  (s)  ^jves  the  meaningless  result  ./-,:  ./;.:./•.:./•,  d  :  o  :  o  :  i). 


2oS  TilKKK   PIMKXSIONAL  CKOMKTIIV 

98.  Poles  and  polars.    Tin-  relation  between  poles  and  polars  may 
be  established   by  means  of  plane  coordinates  as  well  as  by  point 
coordinates.     We  shall  define  the  pole  of  a  plane  >\  with   respect   to 
the  extent  (1  ).  ;i(.'~.  us  the  point  the  coordinates  of  which  are 

P->\  =    •',,'',+  •',-'•:.  +  ••',:,'•«+  -',4'V  ('=    I-     -•    ;!'    ^) 

For  the  case  in  which  A  —  0  the  relation  between  pole  and  polar 
is  the  same  as  that  delined  in  ^  0:2,  as  the  student  mav  easily  prove. 
In  the  eases  in  which  A=0  the  polar  relation  is  something  new. 

The  following  theorems  dualistic  to  those  of  ;<  JH  are  obvious  or 
easily  proved  : 

I.  //'  it  pl/ini'  belongs  t<>  tin1  i'.rt>-nt  it*  }><>/>'  ix  tln>  limit  /mint  in  thr 

plilnt'. 

II.  T"  'in//  plii  n,'  nut  n  i$in<jnlnr  pliin,'  <>f  flu-  I'.rfrnf  <v,,-/v. -.•/«<?/</.••;  '/ 
n/>/'j»i'  ji»/t'. 

III.  T<>  in///  point  cnrrfxpi~>nd»  >i  nn'/<pii'  //"/'//•  /I'/n-n  '///-/  "/////  irJn'n 
tli*    pliiih1  f.rfi'/if  //'/.*>•  //"  sitti/iilitr  pliiiif. 

IV.  A   I'ol,'  //.'.s-  ///   itx  pnl, i r  pl>i »<•  n'lit-n  it//'/  null/  irln  n   f/n-  fxJi/r 
pi, ini'  l»'lnn<i*  to  f//>-  ,.rt,'ni. 

V.  Tin1  pnlf  of  <ni  if  pi i'in i'  />i'x  l/l  nil  xi HIJII]/I /•  p/ii/i,'x  irln  />  xii<-li  r.rixt. 

VI.  It  n  pl< in,'  p  pnxxi'x  thmiii/li  th,1  pnl,'  ,,f  ,i  pl,nn'  i/.  t/i,'//  </  /iitNxrx 
tln'oinjh   tin1  /n,/r  iif  p. 

VII.  All  limit  pointx  lifiii'j  in  it  pliin,-  p  iii-,'  tl/,'  limit  /m/'/itx  ///'  pl<ntt'S 
of  tin-  ,'.it,nt  irliii'li  pnxx  tln-oiii/li  tin1  jinliii'  <;('/>. 

VIII.  /•'"/'   il   xiirfil,;1  of  xi'i'oinl  ,-l,ixx  fl'l't/inllt  xiin/i/liir  pliiinx  it    t'x  /mx- 

xin/f  in  ,i_ii  intinit,'  nunil't'r  "f  ii-ni/x  tn  ,-nnxt i-iK-f  x,'lf-pnl,ir  tfti'iiln'ilrmiK. 

IX.  If  'I    I'm,'    nt    lii'X   i/l    il   pl,l in'  /i.    ilinl   /'   illlil   x   ill',     tin'    pl,iinx   i>t'  tin' 
<'./•//  ///    irliii'li    jiiix*    tli/'nin/li    m,    ilinl   */    ix    tin'    pi/in,'    tlirn/n/li    ,//    iiml    / //>' 
pnl,    "t    n.    tin 'ii   p   ,linl  Y   '//'''   /"/ /•//in/i/i-  cull /  in/ill, -x   /  /   /'   ilinl  x. 

99.  Classification  of  surfaces  of  the  second  class.    The  previous 
sections  enable  us  to  write  the  simplest    forms  to  which  the  equa- 
tion nt  a  surfaee  ot  the  second  class  mav  be  reduced. 

I.  A  -  'I.  Since  the  planes  envelop  a  surface  of  tvpe  I.  ^  l'^. 
we  mav  take  the  results  of  that  section  and  tind  the  plane  conation 
corresjKinding  to  each  tvpe  there.  Consequently,  if  no  account  is 
taken  of  real  values  the  equation  of  the  plane  extent  mav  be 
written  as 


Sl'RFACKS  I)!'  SKCOND  ()]:i>KU  AND  SKCOM)  CLASS    'Jo'.) 

If  the  coel]icients  in  the  original  f(|iiatiiin  are  real  and  the  ori^i- 
n:il  eoiirdinates  arr  also  real,  thru,  by  a  real  eliangu  of  eobnlinates, 
the  equation  lakes  one  or  another  ot  the  forms 

><  i  +  »',  +  "f  -f  «f  —  u, 

"\  +  "•  +  "*•     "4"'—  ^" 

i/  ~  -f-  »'f  --  "'  —  ""'  =  "• 


II.  A  =  0,  but  not  Jill  the  tirst  minors  lire  xero.    \Ve  have  already 
obtained    equation    ('>),    ^  U7,   as   a   possible    ei|iiation    in   this  case. 
li    no  account    is   taken   of   reals   this  equation   can    be   reduced    t<i 

the  form 

a  ~  -f-  ?/.r  -f-  "  j"  :—  ". 

In  the  domain  of  reals  there  are  two  types: 
1.    Planes  tangent   to  a  real  plane  ciirye 

n  ~  -f-  //.f  —  i/  ~  =  i). 

'2.   Planes  tangent  to  an  imaginary  plane  ciirye 
u~+  ?c+  f/.f=  0. 

III.  A  --  0,  all  the  tirst    minors  are   x.ero  but   not    all   the   second 
minors  are   /ero.     As  sho\\n   in   ^  '.'7.  the  equation  can   he  reduced 

to  the  single  type 

n-  i-  n  :,      n 

if  no  account    is  taken   ot    reals,  and   to  the  following  two  types  in 
t  he  domain  of  reals  : 

1  .    Two  real  bund  les  ot      lanes 


CHAPTER   XIV 

TRANSFORMATIONS 

100.  Collineations.    A  collineation  in  spare  is  a  point  transforma- 
tion expressed  by  the  equations 

PJ'\  =  "i  i-r.  +  ".-''.+  "1.7':,  +  "u-'o 


We  shall  eonsidcr  only  the  ease  in  which  the  determinant  !  a.,. 
is  not  /ero,  these  beinj^  the  Hi>nnini/ulttr  eollineations.     Then  to  any 
point  .i\  corresponds  a  point  ./•[,  for  the  I'i^ht-hand  nieinliers  of  (1) 
cannot    sininltaneonslv   vanish.     Also    to   anv    point   ./•'   cori'esponds 
a  point  .r,  inven   bv   the  eipiations  obtained   bv  solving  (1), 

a.r^  .l^  +  .l^+.l^  +  A^,  (-2) 

where,  as  usual,  .\if.  is  the  eofactor  of  a<k  in  the  expansion  of  the 
determinant      ",,.  . 

I>v  means  ot  (  1  )  anv  point  which  lies  on  a  plane  with  coordi- 
nates nt  is  transformed  into  a  point  which  lies  on  a  plane  with 
eoordimiU'S  /'',  where 


..,"'.  +  "..,,":'  4-  ",,".,'•  (  -t  ) 

The  following  theorems,  similar  to  those  of  J^-IO,  may  be  proved 
bv   the  same   methods  there  employed. 

/.   /!//  'i   n'nix/tii/nfii/'  1'nf/iiH'nfi'iii  fi"itifx,  /ifitttix,  it//'/  xfrrtiijJtt   liinn 

il/-'      i  I'll  lixt  nl'lin'fl     t/'f'i    ^'"////.v,     lil'lilifi,     illl'l    xtl'ill'lllt     Itllt'X    I'l'Xpl'I'tH'i'llj    >H 

<i  ciii  -f  •  •-"  //I    itui  a  ii'  /'. 

II.  'Tin    iioHxiii'jiiliii'  i-"l!i  in  iif  iniix  fi>nn  it  i//'"iip. 

III.  It'  /',.    /'..   /',   I\.  <tii<l   /.'  <irt'  //'•<•  <i  rl'it  riirilii  nxminn'il  y/o////.v  H<> 
juiir  ,,t  /rlin'li  in  111  tin'  Kit  /in  i>liim.  iiinl  /,'',  /'/,  /;',  /('',  it/n/  I''  n/'i'  <(lxi) 


TRANSFORMATIONS 


241 


fire  arbitrarily  aswumed  jwintx  >i<>/»t/r  <>f  //•///'•//  //<•  ///  tin  mini*-  /</<//<•, 
there  t'.r/xtx  n>n'  iin<l  "/////  "//c  fnHiiit'ittinn  /"/  /mint*  «/'//7//<7/  /,'  in  //•<//'*- 

fnrinul  into  /;',  /._:  ////"  /:.',  /;  intn  /;',  /;  ////«  /;',  //////  /;  ////.<  /•'. 

IV.  A  H'lnxini/iditr  enUineati»n  entiil'lix/tex  //  /</•<;/'•'•/  /'7///  /.,///•,•«//  ///, 

pntiitx  iif  f  ir<i  1'nrreHfitiHilint/  /•</////»«  "/•  tin'  jiliimx  of  tirn  ivj/vvxy/u///////!/ 
IH'/II-I/K,  uit'l  mij!  sue  /I  fn'iijeefii'i'fi/  nun/  /<»•  fKtitfilix/n-il  in  tin  /'//  //////•• 
number  <>t'  irni/x  />//  <t  nonnhif/nlnf  f<>tlin<  iiti«n. 

V.  An  if   f/i'ii  /'//mix   ii'/iii'/t   enrrex[ininl  /<y  //nii/i*  "/'  '/   H<inxin</n/iii' 
enllitu'ittion  ni'i1  [iroji'i-t  t>'tl  i/  tr<tm*fi>rnte<l  intn  <iirlt  «///»/•. 

101.  Types  of  nonsingular  collineations.  A  cnllincat  inn  has  a 
fixed  [mint  wlicii  r[=r,  in  llic  rtjiiiitions  (_1  ).  ^  l(|l).  l-'i\c(l  jmints 
arc  tht1  re  fore  <j;ivt'ii  liv  the  ('(uutions 


"    ./•  4-^    ./•  4-  </    ./'4-(''     —  p)./1        (>. 

11      1  t'J     'J  |:i     ii     '  II  '     '       I 

Tin1  necessary  and  siiflicient  conditions  tliat  these  e<jtialions 
liave  a  solution  is  that  p  satisties  the  equation 

//  —  p   <i      n       ii 
n   i     u       rt        ii 

</        it  —  o   n        it 

21  -'      =  I). 

n        ii       il  —  p   ii 

:;i          ;)•:         :;:i    r      ;i 

»/          ^/         it          il     p 
ii         rj        i  .;         n   r 

Similar  conditions  hold  for  the  tixed  planes.  l>\  reason  in'_r 
analogous  to  that  used  in  £  11  \ve  may  establish  ihe  results: 

/'Jl'l'l'//    <'n//l  III  iff  In//     //,/N    ,7X    nl'llll/   tlixttnet     //./'I'/    ll/fflll'X    '<••<    fi.l'i'if    /"ill/tfi, 

ilx    ///'/////   ncllt'ilx   nl    fl.i'iil    nlitiiex   <ix   liiii*  ">    f/.ii</   nniiifx,    <in<l  ,/N   iininii 

hunt/leu  "/'  //./>•//  f>/iiiii-x  it*  [il'iinx  nt'  ti.iiJ  /'"'nits. 

Ill  i'fi  /•//  fl.li'i/  fililin  In  ill  li-ilxt  "in'  ti.i'iil  fmhlt  iiml  n>h'  //'./••/  ////., 
tlirnlli/h  ir,/-//  fl.i'ril  ///ii'  i/fi,'*  ill  /,',/*/  i,  H,'  //.iii/  /il'tli'.  nil  i  1'i'i'f/  t'l.liil 
lilli'  //ex  ill  It'ilst  nil,'  //./•<•</  /'"////,  ////-'ill,///  !•/',,'//  tl.i'ii/  /'"lilt  i/n  </f  /,,/*/ 

niii'  ti.ii'il  Inn'  t//ii/  ii/ii'  fi.ri'i/  ii/iiiii'. 

\\'ith     the     aid     of    these     theorems     \\  e     ma\      no\\     ela>>it\     thr 

col  lineal  ions,     l-'oi-  hreyity  \\f  shall  omit    much  of  the  details  nt   t  he 
\\'oi'k,  \vhicli  is  similar  t<>  1  hat    of  vj    II.'      In  t  he  l'ollo\yiii'_;'  ei|  nations 


1M'2  THKKK    DIMKNSloNAI.  <  i  K<  >M  KTR  V 

tin-   letters  '/.  t>.  i:  <f  represent   quantities  which   are  distinct    from 
each  other  ami   from  /.ero. 

.1.  .I'  l>'<t*t  /<inr  tlixt'nft  (isi'il  jHihitu  n"t  in  tin-  x<nn>'  />l<nit'.  The 
four  jmints  mav  l»e  taken  as  the  vertices  of  the  tetrahedron  of 
reference  A  !'>'!>  (see  V\^.  •'>-.  $  SL>  ).  \\'e  have.  then,  the  following 
types: 

Tvi'K   I.  P''[=  "-'V 

p.  ''.',=          /'./•.,, 

P  •'',•-  '••'',- 

p.'\=  '/A- 

The  collineation  has  the  isolated  lixed  points  .1.  /.'.  <  '.  J>,  and  the 
isolated  fixed  planes  M'><\  !;<'/>,  <'I>A.  1>M'>. 

Tvi'K  II.  p.r\=  '/./',. 


The  collineation  has  the  isolated  lixed  points  .1.  /.'.  the  line  of 
lixed  pnims  ('!),  the  isolated  fixed  planes  .1<'I>,  /!<'/>,  and  the 
pencil  of  lixeil  planes  with  axis  AI>. 

Tvi'K  III.  P'\=  «-rv 

.r',-  i  i.r,, 


The  rollineation  has  the  two  lines  of  tixi-d  points  .  )//,  r/>  and 
the  two  jiciicils  of  tixed  planes  with  the  axes  .!/.'.  '  I>. 

TVPK  I  V.  p./-',      "./•.. 

p.r'.  •  '/./',, 

p.r'3  = 

p.r'.   '  '/./',. 

Tin-  collineation  has  the  isolated  tixcd  point  .!.  the  j.lane  of 
lixed  points  !'><  I>.  the  isolated  tix.'d  plane  /!''/>,  and  the  luindle  of 
lixt-d  planes  with  vertex  .!. 


TRANSFORMATIONS 
TVIM:  \r.  p.r\      i/./^, 


P-'\  "•>',- 

All  juiiiits  and  planes  arc  fixed.    It  is  the  identical  transformation. 

/.'.  At  li'/ixf  t/tt'ff  i/ixtitii-f  //./>  </  I'ointu  not  in  tin1  xiiiiif  xtf<ti<//t(  tiin  <///-/ 
it"  "t/ifru  tint  in  tin'  xitiin1  [iliiin  .  1  lie  tixed  points  inav  he  taken  as  the 
jpoints  .1,  /.',  />.  There  are  three  fixed  planes,  one  of  which  is  .!/.'/>, 
and  the  others  must  intersect  At>I>  in  one  ot  the  three  lixed  lines 
.I/.',  <'!>.  I>.\.  \\'e  mav  take  one  of  these  planes  as  I>l'><'(.r  =  0  ). 
Then  in  that  plane  we  have  a  collineatioti  in  which  /.'  and  l>  are 
the  only  fixed  points.  P>v  proper  choice  of  the  vertex  <  '  \  he  coll  in  ca- 
tions in  the  plane  ./•  —^  mav  he  j^'iven  the  forms  found  in  v;  H. 
Ileiice  lor  the  space  collineat  ions  we  find  the  following  tvpes: 

TV  IT:  VI.  p.i\  —  tt.r}  -f    ./-,, 

p  •>''••  —  ''•''.,' 

p.r',  ----  '•./•., 

p.r,  =  tb't. 

The   collineatioti    has   the    isolated    fixed    points   .1.   /.',    I)  and  the 
isolated   tixed    planes   .!///>.  Al><\   /!<'/>. 
TVI'K    \'II.  p.r\  =  it.1-^  +     ./'.,. 


The  collineatioii  has  an  isolated  fixed  point  />.  a  line  of  lixi 
joints  AH.  the  isolated  fixed  plane  .I/.'/',  and  the  pencil  of  tixi 
I  ilaiies  with  t  he  ax  is  ( ' I >. 

TVIM-:  \'l  II.  p,\    --  '(.r  -i-    ./•.,, 

p.i'.,  -  it./  . 

p.r'  ,1.    . 

/»•'-.;  •/.'-,. 

The  collincatioii    has  the   i>olnted  fixed  point     !.   the    line  of  fix. 


214 


TIM;  !•:  i  •:-  1  >  I  M  KN  s  i<  >x  A  i.  <  ;  i  :<  >.M  KTI:  v 


Tvpe  \  III  is  distinguished  e/e<  >iiiet  ricallv  from  Type  VII  l>v  llu1 
fad  that  in  T\pe  V  1  1  1  the  line  of  fixed  [mints  intersects  tin*  axis 
of  tlic  pencil  of  tixcil  planes  and  in  Tvju>  \'II  this  is  not  the  case. 

T\  ri:  IX.  . 


The  rollim-ation  has  the  jilaiie  of  fixed  points  AIll>  and  the 
liiuidle  of  lixed  planes  \villi  vt-rtt-x  1  '  >. 

<  '.  .!/  liiittt  In'"  distinrt  fi.ru  I  i>«nitx  <tn<I  n<>  "tJn'i'x  ii"t  in  tin  x<tnt>' 
>•//•-//.////  I'm:'.  The  lixed  points  mav  lie  taken  as  /.'  and  l>.  'J'liere 
must  lie  t  \\  o  distinct  lixed  planes  of  \vhii-h  one  must  pass  throui^h 
J:I>  and  the  other  mav.  T'here  are  two  siil>ea>es  each  leading  to 
t\\  o  t  \'pes  of  ei  lilineat  i>  'iis. 

1.  It  lioth  lixed  planes  pass  through  HI>  thev  nia\~  he  taken  as 
./  -(l  and  ./•,  ~().  Then  in  each  of  these  planes  we  have  a  eollineatioli 
of  TV  p.-  I  V  nr  TV  pe  Y  of  jj  H  .  \\\  j'l'oper  ehoiee  of  t  he  points  .1  and 
('  we  have,  aeetirdhi'jlv,  the  following  tvpes  of  s[>ai-e  eollineatioiis  : 

TV  IT.     \. 


I  he  ei  ilhneat  ii  ill  lias  i  lie  i- 
ixed  planes  .1  /:/>.  /:<  '/>. 

TVIM:  XI.  o/, 


lated  fixed  points  /.'.  7>and  the  isolated 


p.l  -  ''.I     . 

f.i  I  '/.I      -+-        ./'   . 


Tin-  i  ollineatioii   has  the  line  o|    fixed   point-,  /•'/'  ami   the  pencil 
:    fixed    planes    ",    •  ixis  ///>. 

l'.    I;    onh    one  of  the   fixed  planes  passes  through    /:/>  the  other 
tixed   poims  /.'  or  /'.     In  this  case  we  may 


T  1  :  A  N  S  K<  )  1:  M  ATK  )N  S 

take  the  two  fixed  planes  as  .r  =  0  and  ./•.,  •  0.  Then  in  the  plane 
lie  U  we  have  a  enllineat  inn  nf  Tvpe  I  V  or  Tvpe  V  nf  vj  il  and 
in  AT>1>  one  of  Type  V  I  of  £  II.  liv  pmper  choice  nl  the  points 
C  and  A,  therefore,  \\'e  have  the  following  tvpes: 

TVPK  XII.  ?•*'(=  <'•'•,+    -''.M 


The  eollineatinn  lias  the  li\rd   [mints  //,   l>  and   the  fixed   plan 
licit,  ADC. 

Tvi'K    XIII. 


The  collineat  ion  has  the  line  of  fixed  points  /.'/>  and  the  pencil 
of  fixed  planes  with  the  axis  /)('. 

1>.  (hi///  unc  //./•»•(/  [mint.  The  tixeil  p.nint  nia\'  lie  taken  as  />. 
The  fixed  plane  which  must  exist  niav  lie  taken  as  ./•  =  0.  Then 
in  that  plane  the  eollineation  is  of  Tvpe  \  I,  ^-11.  and  the  pnints 
'''and  /.'  niav  lie  so  chosen  that  the  equations  take  the  form  nf 
Tvpe  VI  there  ^iven.  To  do  this  we  first  select  .r  -  0,  ./•  (I  as  the 
fixed  line  in  the  plane  ./•  =  0.  The  point  .1  mav  lie  taken  as  an\ 
jioint  outside  nf  ./•  =  0.  If  A'  is  the  point  into  \\hich  .1  is  trans- 
formed, the  line  .1.1'  may  lie  taken  as  .r  -  0.  ./-,  ".  Tins  fixes  i  he 
point  //.  Then  C  is  determined,  as  in  Tvpe  \  I,  ^-11.  The  result 
is  t  he  folk  iwimj;  t  vpe  : 

Tvi'K    X  I  \'.  p.r\        (U\  +    •>'.„ 

P  •'''•!       "  '/./'-_,+          -''  .. 

p.r,    -  ,u\  (-   ./r 

p.r\  •'.',. 

The  al»n\e  t  \  pes  exhaust  the  cases  nf  a  iionsiii'4'ular  cnllineation. 
In  a  singular  eollineation  then-  exist  exceptional  pnints,  lines,  or 
planes.  The  disciissimi  of  these  is  left  t«-  the  student. 


240  THKEE   DLMKNSloNAL   GEUMETKV 

EXERCISES 

1.  ( 'oiiMdering  the  translation 

.'•'  —  .'•  -f-  "•     //'=  ,'/  +  ?',     •'-' =  '-  +  '' 
as  a  eollineation,  determine  its   fixed   points  and  the  type  to  which  it 

2.  Considering  the  rotation 

as  a  collineation,  determine  its  fixed   points  and  the  tvpe  to  which   it 

3.  Couriering  the  screw  motion 

a-<  a  eollineation.  determine  its  fixed  points  and  the  tvpe  to  which  it 
belongs. 

4.  Set    up    the    formulas    for    the    singular    eollineation    known    as 
"  painter's  perspective,"  by  which  any  point   /'  is  transformed  into  that 
point  of  a  fixed  plane  //  in  which  the  line  through  /'and  a  fixed  point  < > 
meets  //. 

5.  Find  all  possible  types  of  nonsingular  collineations. 

102.  Correlations.    A   correlation  of  point  and  plane  in  space  is 
defined  by  the  (-([nations 

P"\  ~  "n-'"i  +  ",•-''•_'+  '',:/'';  +  ".i-'V       (  /  =  1,  2,  3,  4  )  (  1  ) 

where  //(  are  plane  coordinates  and  ./•  arc  point  coordinates.  The 
correlation  is  nonsingular  when  nik  ~  •'.  and  we  shall  collider  only 
such  correlations.  Then  anv  point./;  i^  transformed  into  a  definite 
plane  //.  and  anv  plane  /''  is  the  transformed  element  of  a  definite 
point.  MI  that  the  correspondence  of  an  clement  and  its  transformed 
element  is  oiie-to-oiie.  Tlie  points  ./ •_  which  lie  on  a  plane  \\ith 
coordinates  //_  are  transformed  into  plane.-,  n'  \\-liich  pass  through  a 
point  ./•'.  where 

PJt—  -Jn"l+  •',-".+  •',,;".:+  -1,,";-  '  -  ' 

where  ./.;  is  tin-  cofactor  of  ./  .  in  tin-  detci'ininant  it  .  \\'e  mav 
sav.  tlierelore,  that  the  plane  ni  i>  ti'anstornieil  into  the  jiomt  ./','. 
1'oints  \\-liieli  lie  on  a  line  /  are  traiisformrd  into  planes  through  a 
line  /'.  >o  that  we  mav  sav  that  the  line  I  is  transformed  into  the 
line  /'. 


TRANSFORMATIONS 

If  tin-  pi  tint  /'(./-,  )  is  transformed  into  the  plane  y>'(  //  ).  then,  h\  the 

same   operation,  the  plane  y'  is  transformed    into  the  point    /'(.',    ), 
where,  from  (  '1  ). 


Thr   hist   equal  ions  solved  for  //'  c;ive 

P"'         '-     "l,'''['    ^    '',,.'•'•'   +     '';;,•'•''   +    "4,  -''I'- 

'1  he  points  ./•   and  ./'  are   in   general   distinct.     '1  hat   thev  sh 
coincide    it    is    necessar\'    and    sullicient.  as    is    seen    bv   eomparison 
of   (  1  )  and  (  I  ).  that 


(":ll       ^",;)-'',  +  (    ";:,          /3",,  >•''..+  <   ":;:;  -  /3";;;1  )  •'':;+  <  ".;,  ~  P"  ,.,'•'' 

(  '/        p<i    )./•  -f  ( ''      -  P''    ).'•  +  ( ''    —  P'i    )•''  +  ( ''    —  P''    ).'' 

41         i       II  '      1  I-J         '       U4  '      a          '       •!:!        '       ;M/'a    '      '       14         '       41' 

\\'hei'c  p  must  satisfy  the  condition 


il      —  nil  ii  nil  it  nil  //         -  Oil 

11        r     n  ;j       r    -i          ,|;;       r    ;n          -11        r     ii 

in  order  that   equations  (  ~> )  mav  ha\c  a  solution. 

\\  hen  the  coi'irdinales  of  a  point  /'  satisfy  eipiat  iou>  (  .>  ).  it  and 
the  plane  //.  into  which  ii  is  transformed,  form  a  double  pair  ot  the 
convlat  ion.  Since  (  •'•  )  is  of  the  fourt  h  decree  we  see  that  in  general 
a  correlation  has  lour  double  pairs,  but  mav  have  more. 

'I  he  double  pairs  mav  be  made  the  basis  ot  a  classification  ot 
correlations,  as  was  done  in  the  case  ot  the  plane,  but  \\  c  \\ill  not 
take  the  >pace  to  do  so.  (  >l  special  hit  civ  M  is  the  ease  in  \\  hieh  each 
point  n|  >paee  is  a  point  of  a  double  pair.  K«r  this  n  i>  necessary 
and  sulticieiit  that  equations  (.))  .should  be  satislicd  tor  all  values 
ot  ./.  'I  Ins  ean  happen  in  only  two  cases; 

1.   p       1.  -',        .1  (.  -.    p        •  1.  <t        ".  •'.. 

In    the    first    ease    the    correlation    U    evideiith     a    [iolarii\     \\ith 

respect    to    the    collie    V, /../•./•.        d.    and    b\     projier   choice    of    coiirdi- 
nates   it    may  be   represented    b\    the  c(|iiatioiis 


24S  THKEE-IJIMEXSIOXAL  (iEOMETKV 

In  the  second  case  the  correlation   has  the  form 
on'  it   ./•  -4-  <t   ./•  +  a    >• 

r      1  i-j    2    '         l;;'   ;i    '         n"    4' 


P"*=    -  'Vi  -"•;/:•  -'W 

and  represents  a  mill  xtjxtfin,  which  will  he  discussed  later.  It  will 
be  shown  that  bv  choice  of  axes  the  correlation  may  be  reduced 
to  the  standard  form 


P  >'=-•>  3. 

Another  (jiiestion  of  interest  is  to  determine  the  condition  under 
which  a  point  /'  lies  in  the  plane  //.  into  which  it  is  transformed. 
From  equations  (  1  )  it  follows  at  once  that  the  coordinates  of  P 
must  satisfy  the  equation 

V,r,.r,r,=  0. 

This  equation  is  satisfied  identically  only  in  the  case  of  the  null 
system  :  otherwise  it  determines  a  quadric  surface  K^,  the  locus  of 
the  points  /'  which  lie  in  their  respective  transformed  planes. 
Similarly,  the  planes  f>  which  pass  through  their  respective  trans- 
formed points  envelop  the  quadric  A",, 


which  is  in  general  distinct   from   A'f 

EXERCISES 

1.    1'rove  that   if  /'  and  //'  are  a  dmilile  pair  the  plane  j>'  is  the  polar 

plane   "[   /'   with    ropeet   to  the   collie    l\  ^ 

'2.    I'rnve  ihat   a  correlation   is  an  involutorv  transformation  onlv  in 
tin-  ease  of  a  i.olarilv  or  a  null  system. 


TRANSFORMATIONS  iM«) 

103.  The  projective  and  the  metrical  groups.  The  prodm -t  of 
two  nonsingular  collineations  or  <»(  two  nonsingular  correlations 
is  ii  nonsingular  eollineation.  1  lence  the  totality  of  all  collineations 
and  correlations  form  a  group,  since  this  totality  contains  the 
identical  substitution.  1'rojccti^e  geometry  may  be  detined  as  that 
geometry  which  is  concerned  with  the  properties  of  figures  which 
are  invariant  under  the  projcctive  group.  In  this  geometry  the 
plane  at  infinity  has  no  unique  property  distinct  from  those  of 
other  planes  nor  is  the  imaginary  circle  at  intinitv  essentially 
different  from  any  other  conic,  and  all  questions  of  measurement 
disappear.  Quadric  surfaces  are  distinguished  only  by  the  presence 
and  nature  of  their  singular  points. 

Subgroups  exist  in  great  abundance  in  the  group  of  projections. 
For  example,  the  collineat  ions  taken  without  the  correlations  form 
a  subgroup,  but  the  correlations  alone  form  no  group.  All  eolline- 
ations  with  the  same  fixed  points  obviously  form  a  subgroup. 
Again,  all  collineations  which  leave  a  given  quadric  surface  inva- 
riant form  a  subgroup.  Of  great  importance  among  these  latter 
is  the  group  which  leaves  the  imaginary  circle  at  infinity  invariant. 
This  is  the  Metrical  </ruii]t,  which  leaves  angles  invariant  and  multi- 
plies all  distances  by  the  same  constant. 

The  general   form  of  a   transformation   of  the   metrical  group   is 


>'-  A-+//M  +«"'»+  >8f  (1) 

pt'  =  t'. 

where  the  coefficients  satisfy  the  conditions 

/  in   -f-  /  ///  _  -f-  /..///..  =  ///  //   -f-  in  ii   -f-  ///.,//   -~  //  /  -f-  n  J :-{-  n  J   r-  o.     (  :5 ) 

It  is  easy  to  see  that  the  distance  between  two  transformed 
points  is  by  this  transformation  /•  times  the  distance  between  the 
original  points,  where  /.-"  is  the  common  \alue  of  the  expressions 
in  {-),  and,  con\ 'ers(d\',  that  a  eollineation  which  multiplies  all 
distances  by  the  same  constant  is  of  the  form  (1  ).  I  he  preser- 
vation of  angles  follows  from  elementary  theorems  on  similar 
triangles. 


2-iO  THKEE-WMEXSIOXAL  GEOMETRY 

All  iranst'ormations  of  the  metrical  group  which  leave  a  plane/' 
lixed  form  a  group  of  collineations  in  that  plane  by  which  the 
circular  [mints  at  infinity  are  invariant.  This  group  is  therefore 
the  nictrical  group  in  />,  and  the  protective  definitions  of  angle 
and  ilistanee  given  in  £  f)U  stand. 

EXERCISES 

1.  If  /Ms  the  determinant  of  the  coefficients  /,///,//  in  (1),  show  that 
l>  =  ±  /''3. 

'2.  Show  that  the  necessary  and  sutlicicnt  condition  that  (1)  should 
rejireseiit  a  mechanical  motion  is  that.  1>  =  -\-  1.  and  that  it  should  repre- 
sent a  motion  combined  with  a  reflection  on  any  plane  is  that  I)  = —  1. 

3.    Show  that  if  I>  =  ±  1  in  addition  to  conditions  ( L')  and  (.'>),  we  have 

/i"  +  //'i'  +  »i  =  l-<  +  in'-  +  "j1  =  /:f  +  "':f  ~f"  ";iJ  —  1- 

//-    +      "<l"'-;    +      "l"o    =      /./;j    -f     '"./"';,    +     ".^'.j    =     A/;     +      "'./",    4-      "./^     =     0. 

104.  Projective  geometry  on  a.  quadric  surface.  It  has  already 
been  noted  (^  b'U)  that  the  geonietry  on  a  surface  of  second  order 
with  the  use  of  (jiiadriplanar  coordinates  is  dualistic  to  the  geom- 
etry on  the  plane  witli  the  use1  of  tetracyclical  coordinates.  For  in 
each  case  we  have  a  point  defined  by  the  ratios  of  four  quantities 
./•  ,  r  ,  .r. ,  ./•  ,  bound  bv  a  quadratic  relation 

a)  (./•)=(),  (1) 

which  is,  on  the  one  hand,  the  equation  of  the  quadric  surface 
and.  on  the  other  hand,  the  fundamental  relation  connecting  the 
tetracvclical  coordinates. 

Anv  point  /  on  the  quadric  surface  inav  be  taken  as  correspond- 
ing to  the  point  at  infinity  on  the  plane,  since  the  point  at  inlinity 
is  in  no  way  special  in  the  analysis.  Any  linear  equation 

V'V,^0  i'2) 

represents  a  plane  section  of  the  surface  or  a  circle  on  the  plane. 
Should  the  section  pass  through  /,  the  circle  on  the  plane  becomes 
a  straight  line,  but  circles  and  straight  lines  have  no  analytic 
distinction  in  this  geometry. 

If  //,  is  a  point  on  the  qnadrie  surface  and  we  have,  in  (  -  ), 


TRANSFORMATIONS  H-M 

the  plane  ('2)  is  tangent  to  the  surface,  and  the  circle  on  tin-  plant- 
is  a  point  circle.  The  point  ot  tangencv  on  the  snrtace  corresponds 
to  the  center  of  the  point  circle  on  the  plane.  The  intersection  of 
the  tangent  plane  with  the  quadric  surface  consists  of  two  gen- 
erators. In  a  corresponding  manner  the  point  circle  on  the  plane 
consists  of  two  one-dimensional  extents.  Neither  alone,  however, 
can  he  represented  by  a  linear  equation  in  ./-,„  and  therefore  they 
are  not  straight  lines  on  the  plane.  If  this  is  olisciire  it  i>  to  be 
remembered  that  imaginary  Mraight  lines  are  not  defined  liv  any 
geometric  property,  hut  by  an  analytic  equation. 

The  intersection   with  the  quadric  surface  of   the   tangent    plane 

at    /  corresponds  to  the  locus  at   infinity  on   the  plane. 
i 

The  center  //.  of  a  point  circle  on  the  plane,  or  the  point  of  tan- 
gencv of  a  tangent  plane  to  the  surface,  is  found  by  solving  (  '•}  > 
for  //(.  The  values  of  //.  must  satisfy  (1  ).  and  the  substitution 
gives  the  equation  ?(//)=<),  (  1, 

which  is  the  condition  that  a  circle  on  the  plane  with  tet  racydical 
coordinates  should  be  a  point  circle,  or  that  a  plane  in  spare  should 
be  tangent  to  the  point  circle.  It  is  in  fact  simply  the  equation  in 
plane  coordinates  of  the  quadric  surface  (1  ). 

Two  circles  on  the  plane  are  perpendicular  when 


:          ,;, 

^™      I  ll 

In  space  the  pole  of  the  plane  V/r.rt.=  0  with  respect   to  the  sur- 


the condition  that  this  pole  lie  in  the  plane  N  /v/^  0.  lleiice  two 
orthogonal  circles  on  the  plane  with  tetraevdieal  cooi'dinates  cor- 
respond to  two  plane  sections  of  the  quadric  surface  >udi  thai 
each  plane  contains  the  pole  of  the  other. 

A  lineai'  substitution  of  the  tetraevelical  coordinates  corresponds 
to  a  col  lineal  ion  in  space  which  leaves  the  quadric  surface  invariant. 
'I  he  geometry  of  inversion  on  the  plane  is  therefore  duali-tic  to  the 
geometry  on  the  quadric  surface  \\hidi  i>  in\anant  with  respect  to 
colliiieations  which  leave  the  surface  unchanged.  T\\o  points  mi 
the  plane  \\hich  are  inverse  \\ith  respcd  to  a  circle  <  coi-respolid 
to  two  points  on  the  quadric  surface  >uch  (hat  an\  plane  through 


252  THKKK  DIMENSIONAL  (JKOMKTRY 


them  passes  through  the  pole  of  the  plane  corresponding  to  C  or, 
in  other  words,  such  that  the  line  connecting  them  passes  through 
the  pole  of  the  plane  corresponding  to  ('.  Since  the  center  of  a 
circle  on  the  plane  is  the  inverse  of  the  point  at  infinity  with 
respect  to  that  circle,  the  point  on  the  quadric  which  corresponds 
to  the  center  of  a  circle  may  be  found  by  connecting  the  point  / 
with  the  pole  of  the  plane  corresponding  to  the  circle. 

An  inversion  with  respect  to  a  circle  corresponds  in  space  to  a 
collineation  which  transforms  each  point  into  its  inverse  with 
respect  to  a  fixed  plane.  That  is,  if  the  fixed  circle  corresponds  to 
the  intersection  of  the  quadric  with  a  plane  .)/,  and  K  is  the  pole 
of  M.  an  inversion  with  respect  to  M  transforms  any  point  /,'  on 
the  quadric  into  the  point  /.?,  where  the  line  KI[  again  meets  the 
quadric.  The  collineation  which  carries  out  this  transformation 
has  the  plane  M  as  a  plane  of  fixed  points  and  the  point  K  as  a 
point  of  fixed  planes. 

Consider  now  the  parameters  (X,  /j.)  on  the  surface,  defined  as  in 
§  9t>.  They  may  he  taken  as  the  coordinates  of  a  point  on  the  sur- 
face1 and  may  be  interpreted  dualistically  to  the  special  coordinates 
of  vj  70.  The  two  families  of  generators  are  then  dualistic  to  the  two 
s\ 'stems  of  special  lines  of  ^  70,  and  the  locus  at  intinitv  on  the  plane 
is  dualistic  to  the  generators  through  the  point  /  of  the  surface. 

The  bilinear  equation 

f/jX/i  +  ,ia\  +  atn  +  n^  =  0  (6) 

represents  a  plane  section  of  the  quadric  surface  and  is  dualistic 
to  the  equilateral  hyperbola  on  the  plain1  with  two  special  lines  as 
asymptotes.  A  section  of  the  quadric  surface  through  /corresponds 
to  an  ordinary  line  on  the  plane,  from  which  it  is  evident  that  by 
the  use  of  the  special  coordinates  the  straight  line  has  the  properties 
of  the  equilateral  livperbola. 

Any  collineation  of  space  which  leaves  the  quadric  surface  inva- 
riant gives  a  linear  transformation  of  X  and  of  fj..  This  is  evident 
from  the  fact  that  the  collineation  must  transform  the  lines  of  the 
surface  into  themselves  in  a  one-to-one  manner.  It  mav  also  be 
proved  analvticallv  from  the  relations  of  £  (.H;. 

Conversely,  anv  linear  substitution  of  X  and  /j.  corresponds  to  a 
collineation  which  leaves  the  quadric  invariant. 


T 1 1 A N S  K<  >  K  M  A  T  K  > X 8  2 ."> o 

Consider  in  fact  tin-  substitution 
rtV-f/3 

*•=    >r,  *'          /*  =  /*'  (") 

7  A  +  6 

which  l(>ii\'t'S  tin'  generators  of  the  second  family  fixed  and  trans- 
forms the  crfiierators  of  the  tirst  faniilv.  From  (  {  ).  $  '.Mi,  it  is  ea>\ 
to  compute  that  this  is  equivalent  to  the  collincat  ion 

pj\  —  ((i  +  8  ).r(  +  i(«  —  8  )./•',  +  (  7  —  ?}  )./•',  -  /(  tf  +  7  )  j'4, 

p.r,  =  i(—ft  +  8  )./•;  +  (a  +8  )./•'.  4-  '  (  tf  +  7  )•'•'  -f  (  -  tf  +  7  ).'"4', 

,  •         ,  ,    "  ^         -  c-         ;  (  S  ) 

p.r,  =  ( £  —  7  )./-,  —  ?  ( p  -f-  7  )./•,  4-  ( 'i  -f  o  ).'',;  4-  /  (  -  n  +  6  )./r 
p./-4  --  /(  tf  -f  7  )./'!  +  (fi  —  y  ).r',+  i(n  -  6  )/. -f  (  a  4-  ^  )./> 
Similar  results  can  be  obtained  for  the  trunsfornuitioii 

,  //>/u'-f  /' 

X  =  X',  /z  -          ,          •  (  '.»  ) 

/'/*  +  7 

by  which  the  o-eiieratnrs  of  the  first  family  ai'e  fixed,  and  for  the 
product  of  (  7  )  and  ( l>  ). 

Filially,  the  collineation  corresponding  to  the  transformation 

ap'+0  m\'+n 

X=       ,     ->          /*==-         —  (10) 

7/A  +6  ^X  +  Y 

bv  \vbieli  generators  of  the  t\vo  families  are  interchanged,  is  easily 

computed. 

EXERCISES 

1.  Show  that  if  the  quadrir  (1}.  §  (.»<i.  is  the  sphere  ./•i4-  y--r  :.'-=  1. 
the  t  raiisforiuation  A.  =  >  ":'\',  fj,  =  <  '<:'fj.'  rejirociits  a  rotat  ion  of  t  he  sphere 
aliout  the  axis  «'/.  through  an  an^le  c/>. 

2.  Show   that    the   traiisi'onnatioii   A-        /^'.   p-  =• — A'  replaces   each 
point  of  the  sphere  of  Kx.  1   by  its  diametrically  opposite  point. 

3.  (ilitain  a  transformation  of  A.  /i  which   represents  a  general   rota- 
tion of  the  sphere  in  Kx.  1  about  anv  axis  thi'oii^li  its  center. 

105.  Projective  measurement.  The  definition  of  projectiye  meas- 
urement, giyen  in  ^  47  for  the  plane,  can  evidently  be  geiierali/ed  f.  \\- 
space,  and  only  a  concise  statement  of  essentials  is  necessary  here. 

Let  (0(  .r)  =  0  (  1  ) 

be  the  equation  of  any  ipiadric  surface  taken  a>  the  fundamental 
(piadric  for  the  measurement,  and  let 


2/U  THKKK   niMKNSloNAL  CKOMKTHV 

If  .1  and  />  are  any  two  points  and  7't  and  7',  are  the  points  in 
which  the  line  .  1 1>  meets  the  i|iiadric,  then  the  distance  I>  between 
A  and  /.'  is  defined  by  the  equation 

/>-=  Alog(  .I/.'.  7',7'J: 
or  if   >i   and  .:    are  the  coordinates  of   J  and   />'  respect  ivelv, 

(o  ( //. .:' )  -f-  \  r  <w  ( if,  ? )  \  ~  -  r  fo  ( '/ )  i  [  (t)  <  ~ )  i 

/>-  A  log  /;}) 

ft>  ( //.  2  )  —  \  [ft)  (  //.  z  )  J"  —  [(o  ( // )  ]  [(o  (  :  )  J 

Also,  if  <t  and  /<  are  two  planes  and  ^  and  /,  are  the  two  tan- 
LTent  planes  to  the  (piadric  through  the  intersection  of  a  and  />,  the 
angle  $  between  <<  and  /<  is  delined  bv  the  equation 


wln're  >/,  and  r(  are  the  coordinates  of  //  and   f>  rcspeetivelv. 
Two  planes  are  perpendicular  if  each  passes  through  the  p 

the  other:   for.  in  (  4  ),  if  O  (  it.  r  )  —  0.  then  cf>       )  log  (  —  1  )  _-      +  HTT. 

A  line  is  porpeiidic'iilar  to  a  plane  />  if  everv  plane  through  the 
line  is  perpendicular  to  />  :  that  is,  if  the  line  passes  through  the 

pole    ot    /'. 

\Ve  mav  define  the  an^le  between  two  lines  in  the  same  plane 
as  the  angle  between  the  two  planes  through  the  lines  and  perpen- 
dicular to  the  plane  ot  the  lines.  That  is  the  same  as  dclining  the 


rat  io  of  the  two  lines  and  the  two  tangent  lines  drawn  in  t  heir  plane 
to  t  he  i  jiiadric  surface. 

Anv  plane  cuts  the  ijuadric  surface  in  a  conic,  and  the  definition 
of  an^le  and  distance  is  the  same  as  in  the  proactive  measurement 
ot  ^17.  in  which  this  conic  is  the  fundamental  one.  1'rojeetive 
plane  measurement  is  therefore  obtained  bv  a  plane  section  of 

jil'o]eet  IVe    -pace    measurement. 

A  •-  m   Chapter   \  II    we   have  three  cases: 

1.  '!'!>'  /////<.  /•/..,//,•  i;i  a,-.  The  fundamental  ijiiadrie  is  real,  and  we 
consider  onlv  the  space  inside  of  it.  The  -_;vi  iiiiet  rv  in  the  plane  is 
t  he  same  as  in  ^  Is. 


TRANSFORMATIONS 

II.  7V/'     <•///////<•    i-iixi-.     Tin-    fundamental    cjtiadrie    is    ima^inarv. 
The  LTeometrv   in   the  plain-   is  tin-  same  as  in    £  4'.*. 

III.  '/'//'•  jut /-iil'i'l  !••  fiixi-.      I  In-  t  nndanient  al  tjiiadrie   in  plane  coor- 
dinates mav  lie  taken   as 

"I'-f-  "':'  +  »-,  -    ". 

which  is  that  nf  a  plain-  extent  consist  iic_r  of  planes  tangent  to  a 
cniiic  in  the  plane  j-  -  «.  If  this  conic  is  the  circle  at  intinitv.  the 
measurement  becomes  Kndidean. 

If  the  cniiic  is  a  real  circle  at    intinitv.  for  example  the  circle 


>  -—  \ '  ( .'•  —  ./• r  +  ( //  —  // )"—  (  z      z )\ 
and   the  alible   bet\\'een   the   t\\'o  planes 

,u-  4.  I,,/  4-  ,-r  -i-  ,lt  —  I)     and     (/'./•  +  l-'/i  -1-  <•':•:  4-  •/'/  =  " 


Tlirough  anv  point  in  space  LT<»CS  a  real  cone,  such  that  the  dis- 
tances from  its  vertex  to  points  inside  it  are  imaginary,  distances 
from  its  vertex  to  points  oin>ide  it  are  real,  and  distances  from  its 
vertex  to  points  on  it  are  /.em.  Anv  plane  section  thnm^h  the 
vertex  is  divided  into  regions  with  the  properties  described  in  vj  .»'(. 

106.  Clifford  parallels.  \Vhen  a  system  of  pmjective  measure- 
ment has  been  established,  the  concept  of  parallel  lines  max  be 
ii it  rod i iced  b\-  adopt  in^  >oine  jiropeii  \  of  parallel  line--  in  Kuelidean 
•j;eometrv  as  a  definition.  IVrhaps  the  most  ob\  imis  as  \\ell  as  the 
most  common  deiinitioii  is  that  parallel  lines  are  those  which  in- 
tersect at  intinitv.  \\\  this  deiinitioii.  in  [larabolie  space  one  and 
onlv  one  line  can  be  drawn  through  a  point  parallel  to  a  LMVCII  line, 
in  hvperbolie  space  tv/o  such  parallels  can  be  diawn,  and  in  elliptic 
space  no  I'eal  parallel  can  be  drawn. 

In  elliptic  space,  however,  there  exist  certain  real  lines  called 
( '//tl",-il  /"'/•<///<  /x  which  ha  ve  other  properties  of  parallel  lines  as  they 
exi>t  in  Knclideaii  space.  \Ve  \\ill  proceed  to  discuss  these  lines. 

\\  e  ha\'e  seen  that  anv  linear  transformation  of  the  parameters 
X  ami  u  which  define  a  point  on  a  i|iiadric  >nrt'ace  correspond  to 


12")6  THKKK   DIMENSIONAL  GEOMETRY 

;i  eollineation   which   leaves  tlui  quadric   invariant.    Among   tliese 
transformations  arc  those  of  the  type 


X  =  -  ,          u  =  /x  ,  (1) 

y\'+B 

wliieli  transform  the  generators  of  the  first  family  among  themselves 
lint  leave  each  generator  of  the  second  family  unchanged. 

For  reasons  to  he  given  later  we  call  such  a  transformation  a 

fr'i/iti/'iti"ii  "f  t/n'  tirxf  kin<1. 
Similarly,  the  transformation 

,       N/  nifi'+n  , 

X  =  X,          n  =  -,  (2) 

]>t*  +  y 

hv  which  the  generators  of  the  second  family  are  transformed  lint 
each  of  the  first  family  is  left  unchanged,  is  called  a  translation  of 
f/if  iH'cnnd  kntil. 

Consider  a  translation  of  the  first  kind.  On  the  fundamental 
quadric  any  generator  of  the  second  family  is  left  unchanged  as  a 
whole,  hut  its  individual  points  are  transformed,  except  two  fixed 

points,  for  which  rtX-f/3 

X  =  -  -  -       •  (  d  ) 

7  A.  +  6 

This  equation  defines  two  generators  of  the  first  kind,  all  of 
whose  points  are  fixed.  Hence,  ///  <t  translation  <>f  t/ie  first  kind  (Jn-rr 
itri',  in  i/i'tiiTtil,  t>r<>  <j<'nt'r<tt<n'x  <>f  tin'  frrxt  kin<l  ir/ii<'/i  tire  fu'>'<l  j>t>int 
l>ii  />"/'/!/.  \Vc  say  "in  general"  hccause  it  is  possible  that  the  two 
n  nits  of  (  '•}  )  may  be  equal. 

Call  the  two  fixed  generators;/  and  /i.  Then  any  line  which  in- 
tersects //  and  h  is  fixed,  since  two  of  its  points  are  fixed.  Also 
through  any  point  /'  in  space  one  and  only  one  line  can  be  drawn 
intersecting  _//  and  //.  Therefore,  ti/ii/  />/>i/if  I'  is  t  ran  ft  formed  into 

illf'tlnT  jmiiit  "II  t/ii'  I!  Hi'  /r/i/'r/i  y/f/.v,y,  >•   f/irnH'/Jl    /'  dllif  inti'MCf'tS  </  ilU<l  It. 

Since  we  are  dealing  with  a  ease  of  elliptic  measurement  the  lines 
//  and  //  arc  imaginary.  Then,  if  a  real  point  /'  is  transformed  into 
another  real  point,  the  roots  of  (  :»  )  must  be  conjugate  imaginary, 
since  a  real  line  intersects  an  imaginary  quadric  whose  equation  has 
real  i  -oc  t  tii  -it  -i  n  s  in  con  jugate  imaginary  point  s  corresponding  to  con- 
jngate  imaginary  values  of  X  and  p..  Therefore,  if  ii  tranxlation  <*J  ///'• 

///">•/    /'///«/   ff(l)IKf'nr//IK    l''-'ll   ]>'>intx    illtu    fi'ilf   fin/'/ifK,    f//i  I'*'   iilllxt   In'    tll'n   iJlX- 

ti/n-t  (j.i'1'i/  tff'n<'ra(ori<  f'>/-/-,  ni><>n<lin<i  f"  I'ni)  iui/iit,'  in\injinarif  valum  ol  X. 


TRANSFORMATIONS 

This  mav  also  be  established  by  equations  (  S  ),  s  1<>4.  That  the>e 
may  represent  ;i  real  substitution  8  must  In-  conjugate  imaginary  in 
cr,  and  7  conjugate  imaginary  to  —  /^.  NVe  therefore  place  a  =  </  4-  //•. 
B  —  il  —  if,  ft  —  —  /»  +  /''.  7  =  /'  -f-  /<',  and  have 


,  •       ,    ,  ' 

p.r.  =  -  A./',  4-  iu:,  +  '/./;..  +  f./-4, 

PV  •  -".'-,'-  ^  -  '••'•'»  +  '/.'•;. 

With  these  values  of  n.  rf,  7,  and  6  the  mnts  nf  (  :!  )  arc  con  jugate 
imaginary. 

To  lind  I  he  projective  distance  bctu'een  a  point  ./,  and  its  trans- 
formed point  ./•',  \\'e  use  equations  (  4  )  and  substitute  in  ('•'>).  $  !<•.">, 

placing  I\  -     •     There  results 


r/  +  /\  t+  ,-+  r-  ,/ 

I)  =       lt>g-  =  cos  — -i 

-  (/  —   /  N    //--f   //-+   <•'  \    lf+   /---f-   r---f-  ,/- 

•\vhich  is  a  constant.     Hence,  /<//  //  trtnixfiftittn  <>/  tin-  ilrat  /•///</  »'i/-7/ 

Jutint    nf  i<l>H,'i'    /X    Hl»rt'<?    tln'nUijh     it    fiinxt'lllt     llfijt'Ctil'l'    i/ixtil/lff    "/I    f/i> 

struii/Jtt  littf  ir/iifJi  j>tif<n,'x  tJirniiifh  tin'  [><>int  <tn<l  ///»--Vx  ///,•  ///•//  //./•»-/ 
ili'n*'t'<tt<n'x  <>n  tin-  fundiinn'ntnl  umtdric. 

Ii  is  this  j»ropcrtv  which  gives  to  the  transformation  the  name 
"translation"  and  to  the  lines  which  intersect  the  l\vo  ti\ed  gen- 
erators the  name  "parallels."  l>v  the  transformation  ihe  points  of 
space  are  moved  along  the  Clifford  parallels  in  a  manner  analo- 
gous to  that  in  which  points  are  moved  alon<_>-  Kudidean  parallels 
bv  a  I^uclidean  translation. 

In  the  pi'o  ject  i\  c  space  a  dualistic  pi'ojiertv  exists.  Since  the 
Clifford  parallels  ai'e  fixed,  anv  plane  through  one  ol  them  is  irans- 
formed  into  another  plane  through  it.  Now  anv  plane  contain^  one 
Clifford  parallel,  since  it  intersects  each  nf  the  tixed  generators  in 
one  point.  It  //(  and  //'  are  the  oi'i(_;'inal  and  the  transformed  plane 
respectively,  the  an^le  between  them  is,  bv  (  4  ).  s  1(|"», 


258  THREE   DIMENSIONAL  GEOMETRY 

Hence,  }•!/  '/  translation  <>f  the  first  kind  I'/tfJi  j>t<tnt'  <>f  ttpmr  As-  turned 
<il»i)it  fix'  ('lirt'ord  }><ir<itl'/  in  it  throw/ft  <i  <'<>nxt<int  nn</t<'  tt'hic/i  in  e^ual 
t<>  t/ii'  dixtttHi't'  t//r<»i;//i  ii'tiirfi  jinhttx  i if  the  x/xti'c  <tr>'  »i»r<'d. 

Similar  theorems  hold  for  translations  of  the  second  kind.  The 
two  kinds  of  translations  differ,  however,  in  the  sense  in  which  the 
turning  of  the  planes  takes  place. 

I)V  a  translation  of  the  second  kind  Clifford  parallels  of  the  iirst 
kind  are  transformed  into  themselves.  For  l>y  the  translation  of 
the  second  kind  all  generators  of  the  iirst  kind  are  fixed,  and  conse- 
quently any  line  intersecting  two  such  generators  is  transformed  into 
a  line  intersecting  the  same  two  generators.  Hence  dm  Clifford 
fHtrallt'lx  iD'i'  t'rcri/irJn'rc-  fijitidintant  if  the  distance  ix  measured  on 
Clifford  i>ar<t/lclx  <>f  tin-  nt/,.-r  kind. 

Let  Ll\  and  J/.Y  be  two  Clifford  parallels  of  the  first  kind,  // 
and  //  the  two  tixed  generators  which  determine  the  parallels,  and 
I'(t)  any  line  intersecting  both  Ll\  and  J/.Y.  The  line  /'(,>  intersects 
two  generators  //'  and  //'  of  the  second  kind  and  is  therefore  one 
of  a  set  of  Clifford  parallels  of  the  second  kind.  Therefore  there 
exists  a  transformation  of  the  second  kind  by  which  l'(t)  is  fixed 
and  l.I\  is  transformed  into  J/A',  /'  falling  on  (t>.  Hence  the 
angles  under  which  /'',>  cuts  Ll\  and  J/.Y  are  equal,  of  course  in 
the  project  ive  sense.  That  is,  if  <i  l/'/it-  citf*  t/r<>  <_'/i_tt'»r</  puralli'h, 
f/if  1'nrri'npnndinij  angles  <(/•<•  >'<[\ml. 

In  particular  the  line  may  be  so  drawn  as  to  make  the  angle 
/./'(,>  a  right  angle.  For  if  (J  is  on  J/.Y.  the  point  (t>  and  the  line 
L K  determine  a  plane  ji,  and  in  this  plane  a  perpendicular  can  be 
drawn  from  (t)  to  LK.  To  do  this  it  is  onlv  necessarv  to  connect  (,) 
with  the  point  in  which  the  plane  /'  is  met  bv  the  1'cciprocal  polar 
of  I.l\  with  I'espect  to  the  (piadric  surface. 

Ildice.  /'/•'///>  iinif  p</int  in  "in1  nl'f/rn  ('/itj'n/'d  fi<n'ii//i-/x  <i  fnnirnnH 
i/i'r/» 'ndi<-iil<ir  I'ltn  In-  ilrnirn  in  ///,•  tim,  and  f/n'  >>i>rtini>  <>f  t/tr  fn'riirn- 
il/nilii/-  ini'llldi'd  Iff n-fi'ii  tin-  tiru  j><trn//i'/x  ix  of  ru/ixfi/t/f  lt'»</f//. 

107.  Contact  transformations.  A  transformation  in  space,  expres- 
>ible  bv  means  of  analytic  relations  between  the  coordinates  of 
point-,  mav  be  of  three  kinds  according  as  points  are  transformed 
into  points,  surfaces,  or  curves  respectively.  \Ye  shall  lind  it  con- 
venient to  employ  ('artesian  coordinates  in  discussing  these  trans- 
formations ;md  to  introduce  the  concept  of  a  plane  element. 


TRANSFORMATIONS  l^.V.I 

Let  (.>:  //.  2)  be  a  point  in  spurt-  and  let  /,  —  z  —  /<  (  A'  —  j- )  +  y  (  }       // ) 

he  ;i  plane  through  it.  Then  tin-  live  variables  ( ./•,  //.  :.  /«,  //>  define 
;i  j>lniti'  i/i'/mnf,  which  niiiv  be  visuali/.ed  as  an  infinitesimal  portion 
of  a  plane  surrounding  a  point.  In  fact,  not  the  magnitude  of  tin- 
plane  but  simply  its  orientation  conies  into  question,  just  as,  in 
fixing  a  point,  position  and  not  magnitude  is  considered.  If  anv 
one  of  the  live  elements  is  complex,  then  the  plane  element  is 
siniplv  a  name  for  the  set  of  variables  (>,  //.  r.  y,  y ). 

Since  the  live  variables  are  independent,  there  are  s^'  plane  ele- 
ments in  space.  Of  chief  interest,  however,  are  two-dimensional 
extents  of  plane  elements.  Such  an  extent  we  shall  denote  bv  J7, 
and  shall  consider  three  types: 

1.  Let  the  points  of  the  plane  elements  be  taken  in  the  surface 
2— /'(.r,  // )  and  let  ]>  and  y  be  determined  bv  the  equations 

/>  —      •    '/ —     "•'      More    generally,    let    ./•,    //.    and    .r    be    defined    as 

CJ-  (I/ 

functions  of  two  variables  u  and  r,  and   let  j>  and  y  be  determined 

bv  the  equation 

dz-piU  +  tnly  (1) 

for  all  differentials  <lt(  and  *//•.    Then 


whence  y<  and  y  are  also  determined  as  functions  of  /'  and  '•. 

In  either  definition  the  J/.,  consists  of  the  plane  elements 
formed  b\-  the  points  of  a  surface  and  the  tangent  planes  at 
those  points. 

l!.  Let  the  points  of  the  plane  elements  be  taken  as  functions 
of  a  single  variable  n  and  let  />  and  </  be  a^ain  determined  hv 

O  i  I  - 

equation  (1).  where  one  of  the  t  \\  o  (sav  /-')  i-  arbitrary  and  the 
other  (  sav  y)  is  thus  determined  in  terms  of  y.  and  //.  The  .!/ 
then  consists  of  the  point-  of  a  curve  and  the  tangent  plane-  to 
the  curve  at  those  point-.  The  point-  themselves  torni  a  one- 
dimensional  extent,  and  throiigli  cadi  jmint  <_TOC>  a  ,,]ic-dinieiisional 
extent  of  plane-  :  nanieh  .  t  he  pencil  of  plane-  i  hroii^h  1  he  tangent 
line  t<  i  the  ciir\  e. 


•Jtil)  THKKK   DIMENSIONAL  (JKOMKTHY 

:'..  Let  (./',//,  -")  be  ;i  tixcd  point  and  let  /<  and  7  be  arbitrary  and 
independent.  I  lie  .I/  then  consists  of  a  point  with  tlie  bundle  of 
planes  through  it.  In  this  ease,  also,  equation  (  1  )  is  true,  since 
•  /./-.  <///.  and  >lz  are  all  /.ero. 

It  i>  clear  that  the  .l/.,'s  deiined  above  do  not  exhaust  all  pos- 
sible types  cif  two-dimensional  extents  of  plane  elements.  For 
example,  we  mi^ht  take  the  points  as  points  on  a  surface  and  the 
planes  as  uniquely  determined  at  each  point  but  not  tangent  to 
the  -urfaee:  and  other  examples  will  occur  to  the  student.  The 
above-mentioned  types  exhaust  all  cases,  however,  for  which  equa- 
tion (  1  )  is  true,  as  the  student  may  yerify.  We  shall  say  that  a  set 
of  plane  elements  satisfying  (1  )  form  a  union  of  elements. 

Two  .)/ ,'s  are  said  to  be  in  contact  when  they  ha\'e  a  plane 
element  in  common.  From  this  definition  two  surfaces,  or  a  curve 
and  a  surface,  are  in  contact  when  they  are  tangent  in  the  ordi- 
nary sense,  a  point  is  in  contact  with  a  surface  or  a  curve  when 
it  lies  on  the  surface  or  the  curve,  two  curves  are  in  contact  when 
they  intersect,  and  two  points  are  in  contact  when  they  coincide. 

A  contact  transformation  is  a  transformation  by  which  two  M  's 
in  contact  are  transformed  into  two  J/".,'s  in  contact.  There  are 
thiee  types  of  such  transformations,  which  we  shall  proceed  to 
discuss  in  the  following  sections. 

108.  Point-point  transformations.  This  transformation  is  deiined 
by  three  equations  of  the  form 


O) 


or,  mure  generally,      /•'l(./.  //.  :.  ./ '.  //',  z' )  —  ". 

/•'(./.  //.  :.  ./'.  //'.  .:'  )       u,  (-1) 

/•'  (  .'.  //.  z,  /.  //'.  .;•'  )       I), 

u  here  we  make  the  hypothesis  that  equations  (1  )  can  be  solved 
for  ./.  :i,  2  and  equations  cl\  \«r  ./.  _//.  .:  and  ./-',  //',  z'.  and  that  all 
hint-linns  arc  cunt  mumis  and  may  be  differentiated.  \\  itlnn  a  prop- 
erly  restricted  region  the  n-latinns  liet  \ycen  ./.  //.  ^  and  ./'',  //',  z'  are 
"!!••  to  one,  a  piiint  Lr"(>>  i"'"  ;l  point,  a  surface  into  a  surface, 
and  ,i  <  u  :  \  .•  mt  <  <  a  >  •  ,  \  •  <•. 


TRANSFORMATIONS 


lilil 


A  direction  </./•  :  '///  :  «/:  is  t  rans  formed  into  a  direction  </./•':  <///':  </.:', 

where  /  ,  / 

I     /          '  •'  '      ;  '"•''      ;  '  •'  '      j 

•  /./•'  ,/./'  +          ,///  +        -r/2, 

•  '          • 


7       '  ' 

dy  -    •• 


(  ./'  i  I/  (  Z 

From  this  it  follows  that   two   tangent   surfaces  are  transformed 
into  tangent  surfaces.     More  specifically,  the  relation 

which  defines  a  union  of  line  elements,  i>  transformed  into 
cz'          f  z'      i  z'  (2 

dz'       -+/'  +<r 

c.r  cz        (i/  cz 


.*  t  .*  '    .<  >    .» 

+  /'  -  +  v         =0. 

1  '          (if  c 


( ./'  cz       (>/  cz 

If  now  we  define  ft'  and  //'  so  that  this  relation  is 


a  union  of  plane  elements  (  ./•.  //,  ,r,  /»,  </  )  is  transformed  into  a  union 
of  plane  elements  (./•',  //',  .z',  //,  >/'  ).     l-'rom   equations  (  •>  )   and   (•>), 

p'=t\(j\  .'/•  z,  /'-  '/>• 
'/'=./'.  (./'.  //.  ,r,  /-.  7  ). 

These  e(|uations  adjoined  to  (1  )  form,  together  \\ith  (1).  the 
t'nldt'i/i'il  fiiimt  tritnxfi.ifiHittitin*. 

A  collineation  is  an  example  of  a  point  trunsi'ormation.  Another 
example  of  importance  is  the  transformation  l>v  I'eciprocal  radius. 
or  invfi'sion  with  I'cspeet  to  a  sphere.  If  the  sphere  has  its  center 
at  the  oriin  and  radius  //,  the  transformation  is 


202  THREE-DIMENSIONAL  r.KOMKTRY 

EXERCISE 

IMscuss  the  properties  of  the  inversion  with  respect  to  a  sphere, 
espeeiallv  with  reference  to  singular  points  and  lines. 

109.  Point-surface  transformations.  Such  a  transformation  is 
defined  by  the  equation 

/(.,;  //,  z,  .1'.  //',  z')=0,  (1) 

\\itli  the  usual  hypotheses  of  continuity  and  differentiability  of/ 
An  example  is  a  correlation  since  it  may  be  expressed  by  the  single 
equation 

C'V'+    "l3#+   "l/    ^   "ll  )•'''+<  ",!'''    +",,//   +   ",/   +",,)// 

-f-  ( ii ..''-(- '/,//-)-''.,. .?  -f-  ''.   )  -  4~ ''  •''  +  ''  ,y  +  ''.,-  4~ ''    =  0. 

l>y  eijuation  (  1  ),  if  ( .r,  //,  ^ )  is  tixed,  ( ./•',  //'.  z  )  lies  on  a  surface  ///', 
and  we  say  a  point  /' is  transformed  into  a  surface  >//.  If  /''(./•',//',  z' ) 
is  tixed,  the  point  ( ./•,  if.  z'~)  describes  a  sui'face  ///,  where  the  surfaces 
///'  and  m  are  not  necessarily  of  the  same  character.  It  /''  is  on  ///'  it  is 
ob\  ions  that  in  contains  /'.  In  other  words,  if  /'  describes  a  surface 
///.  the  corresponding  surface,  in',  continues  to  pass  through  I'1.  \Ye 
say,  therefore,  that  the  surface  m  is  transformed  into  a  point  /''. 

If  /'describes  any  surface  N  (differing  from  an  ///  surface),  the 
sui'face  in'  will  in  general  envelop  a  surface  .s",  the  transformed 
surface  of  S.  Analytically,  from  the  general  theory  of  envelopes,  if 
the  equation  of  ,s'  is  _.  _  .,  .  > 

and  {>  —    — i  iy  =    —•>  the  tjquation  of  S'  is  found  bv  eliminating  .r,  i/, 
ex  <  if 

and  z  from  (1)  and  ('!)  and  the  two  equations 

-+/'-  =  °<  (3) 

(4) 

Furthermore,  the  tangent  plane  to  .s1'  at  anv  point  is  the  same  as 
the  tangent  plane  to  ///'  at  that  point,  and  hence,  if  we  use  //  and  'j'  to 
lix  that  plane,  we  have 

(O) 


TRANSFORMATIONS  •_!(;;] 

\\  e  now  have  live  equations,  namelv  (  1  ),  (  ;>  ).  (  4  ),  (  •">  ),  and  (  ti  ). 
establishing  a  relation  bet  ween  a  plane  element  (  j;  //,  ~,  //,  7)  and 
a  plane  element  (./•'.  //.  £',/>',  7').  These  equations  niav  l»e  solved 

to  obtain  the  form 

./•  =  (^(.r,  y.  2,  />.  7), 


z=  <t>.(.r,  //.  ,?.  y.  7), 
//-(/>,(./-,  //,  ?.  /*.  7), 
</-  0.(./'.  //,  ~.  y.  7). 
which  form  the  enlarged  puint-surfai-u  eontac-t  transformation. 

EXERCISES 

1.  Studv  the  ti'ansfornuition  detined  l>v  the  e(|iiatiou 

.<•-  -f  .//-  +  :.'2  -  (SJT*  +  ;/;/'  +  :.:.')  =  0. 

2.  Study  the  transformation  detined  hy  the  equation 

(x  —  j-'r  +  (//  —  //')-  +  (::  —  ,-.')-  =  </-. 

110.  Point-curve  transformations.  (\»nsider  a  transformation 
defined  bv  the  t\Vo  L'qiUltiollS 

/',(./'.   if.  z.  /.   if,  z  )=  (), 

(1) 

./.,(./'.  //,  ,?,  ./•'.  //'.  ,?')=  0. 

If  a  [>oint  /'(•''•  //•  -i'  )  is  lixed,  the  locus  of  /''(./',  //',  ,:'')  is  a 
cnr\e  /r'  detined  b\  iMjiuitions  (1  ).  Similarly,  if  /''  is  fixed,  the 
locus  of  /'  is  a  curve  /.'.  Ileiiee  the  transformation  ehanu'es  points 
into  ciii'ves. 

If  /'  describes  a  curve  ('.  the  ciir\c  /•'  takes  ~s_l  positions  and  in 
^eiiei'al  generates  a  surface.  The  s^  curves  /•'  ma\.  ho\\c\fi'.  ha\  e 
an  envelope  C\  which  is  then  the  transformed  curve  of  (  '.  Or, 
linallv.  if  ('  is  a  curve  /•.  the  corresponding  curves  /  '  pa>s  throii!_;h 
a  point  /''.  \\hich  we  have  seen  to  correspond  to  /•. 

If  the  point  /'  describes  a  surface  .V,  the  corresponding  curves  // 
forma  two-paranu-ter  family  ot  curves.  The  envelope  of  the  familv 
is  a  surface  ,s''  \\hich  corresponds  to  N. 

To  work  anahlicallv  let   us  foi'in  from  (  1  »  the  ctiuation 


2G4  Tl I II EE-DI M ENSIGN AL  (iE( )M ETR Y 

With  (y'.  //',  z' )  fixed.  (  -)  represents  a  pencil  of  surfaces  through 
a  A'-curve,  and  the  tangent  plant1  to  any  one  of  these  surfaces  at  a 
point  on  the  /--curve  has  a  />  and  a  7  given  hy  the  equations 

C  f  C  f  Cf  (  f 

•  i  +  \   :-a  ~J  +  \  i  i 

™ f*  CJ_ CJ_  ,ON 

/'  = r'  (J  = : r'  (") 

df          cf  ct       N  cf 

•  i  +  X  '-5  '  '  +  X  -  -J 
cz           cz                         cz  cz 

There  is  therefore  tluis  defined  a  pencil  of  plane  elements  through 
a  point  1'  and  tangent  to  a  A--curve  through  that  point. 

Similarly,  with  (r,  //,  z)  tixed,  equation  ( '2 )  defines  a  pencil  of 
surfaces  through  a  /"'-curve,  and  a  corresponding  pencil  of  plane 
elements  is  defined  by  (r',  //',  z' )  and 


c.r 


ct  r/^, 

•     +X-- 

C2  C^' 


From  (  •_}  )  and  (4)  it  is  easy  to  compute  that  dz—pdf  —  qdy  is 
transformed  into  dz'  —  p'dx1  —  q'dy'  except  for  a  factor.  So  that  if 
(./•,  //,  z,  /i,  (/)  is  transformed  into  (./•',  //',  z',  j>\  >f  )  by  means  of  (  1  ), 
(  o  ),  and  (4),  a  union  of  plane  elements  is  transformed  into  a 
union  ol  plane  elements. 

From  the  six  equations  (1),  (o),  (4)  we  may  eliminate  X  and 
obtain  live  equations  which  may  be  reduced  to  the  form 


</'=/,(•>'<  /A  ?<  /'.  '/)' 
2'  =  /„(/-,  /A  .*,  /',  7), 
/>'=/4(A  //,  »',  y*.  7), 

'/'  =./'.(./',    //,    2,   //.    7), 

which    define    the    enlarged    point-curve    contact     transformation 
derived    from   (  1  ). 

Consider  a  fixed  point  /'(</.  f>.  <•)  with  the  J7,  of  plane  elements 
through  it.  Equations  (1  )  dcline  a  /r'-curve.  and  we  may  consider 
them  solved  tor  z  and  //'  in  terms  of  ./•'.  In  (:))  />  and  7  mav  be 
taken  arliitrarily.  Then,  if  the  values  of.?'  and  //'  in  terms  of  jr'  are 
substituted  in  (•}).  both  X  and  j-'  mav  be  determined.  Finally, 


TRANSFORMATIONS  l^f) 

p'  and  if'  are  determined  from  (4).  This  shows  that  a  definite 
plane  element  through  /'  is  triinsfonned  into  a  definite  jilane  ele- 
int-nt  of  a  /.•'-curve.  The  M  through  /'  is  therefore  transformed 
into  a  .!/.,  along  /•'. 

A  pencil  ol  plane  elements  through  /'  will  in  general  be  trans- 
formed into  un  .)/  ot  plane  elements  lonnin^  a  strip  along  //,  but 
if  the  axis  of  the  pencil  through  /'  is  tangent  to  a  /.--curve,  the 
peiieil  will  l>e  transformed  into  a  similar  pencil  at  a  point  of  the 
//-curve. 

That  being  established,  we  see  that  if  <'  is  anv  curve,  and  \s  e 
take  an  .)/,  of  plane  elements  tangent  to  it,  we  shall  have  corre- 
spondingly an  .)/',  of  plane  elements  forming  a  sui'faee.  lint  if  (' 
is  the  envelope  ot  /.--curves,  the  .!/'  consists  ot  elements  tangent  to 
a  curve  (''  ell\'elopcd  b\'  /.-'-clll'N'es. 

If  /'  describes  a  surface  ,S',  and  we  take  the  .I/.,  of  tangent  ele- 
ments, we  shall  have  a  corresponding  •'/.,,  forming  a  surface  .s''. 
A  plane  element  of  the  .17,  gives  a  delinite  plane  element  ot  a 
/.•-curve,  as  we  have  shown.  Therefore  the  surface  N'  is  made 
of  plane  elements  belonging  to  //-curves  and  is  the  envelope  ot 
such  curves. 

EXERCISE 

Study  in  detail  the  transformation  defined  by  the  equations 

(./•'-I-  ///')  -  :.';:  —  .'•  =  0, 
,-:(.'•'    -  ///')  j-  ,-.'—  //  —  0. 


riLUTKli   XV 

THE   SPHERE  IN  CARTESIAN  COORDINATES 
111.  Pencils  of  spheres.    Tin-  equation 


0  (  1  ) 

i   i         i- 
and  the  radius  r, 


If  </  --  il.  equation  (  1  )  represents  a  plane  which  mav  be  regarded 
as  a  sjiliere  with  an  intinite  radius  and  with  its  center  at  intinitv. 

For  con veiiieiiee  we  shall  denote  the  left-hand  member  of  equation 
(  1  )  bv  N.  The  equation 

shall  then  denote  the  sphere  with  the  coefficients  '',./,.  ,'/,.  /<,,  t\. 
( 'oiisidcr  now  t\\"o  spheres 

N  =  il.          ,s\-=  i).  ( :i ) 

Thev  intersect  at  ri^'ht  angles  when  and  onlv  when  the  square 
of  the  distance  between  their  centers  is  equal  to  the  sum  of  the 
squares  of  their  radii.  The  condition  for  this  is  easilv  found  to  be 

The   spheres  defined    bv   the  equation 

S}+  \S        (I.  (o) 

\\heiv  X  i^  an  arbitrary  parameter,  torm  a  /><  n--//  ot   spheres.     It    >', 

and   >'.,  are  Imth  plane-,  all  sphei'es  of  the  pencil  are  planes.    <);hcr- 
wi>e  the   peiii-il    ci  in1,  am-   otic   and    onl\    one    plane,   the   equation   <>t 

\\  hich   i-   toiind   li\    |  .laciii'_;'  X  '    in  (  "» ). 


TIIK   SIMIKKK    IN    <  AKTKS1AN    C<  M  >K1>IN  ATMS         21)7 
The  centers  of  the  spheres  of  the  pencil   h;i\e  the  coimlinates 

/     _  /I    +    V'-,  //,+    \''^  /'1   +    X/':\ 

\       »/,+  \'i..          ./,  J-  \//,          '/,+  X'/J 

and    therefore    lie    in    a    straight    line    perpendicular   to   the    radical 
plane.    This  line  is  the  /fur  <//<•,///,  /-x  of  the  pencil. 

\\"e  ha\'e  three  forms  of  a  pencil  of  real  spheres  not  planes  : 

1.  When  the  spheres  A'  and  N,  intersect  in  thermic  real  circle  '  '. 
The  pencil  consists  of  all  spheres  through  <  '.  The  radical  plane  is 
the  plane  of  <  '.  and  the  line  of  centers  is  perpendicular  to  t  hat  plane 
at  the  center  (if  <  '. 

'1.  When  the  spheres  ,s'  and  ,s',  intersect  in  an  imaginary  circle. 
All  spheres  of  the  pencil  pass  through  the  >ame  imaginary  circle, 
hut  in  the  ordinary  sense  the  spheres  do  not  intersect.  The  radical 
plane  is  a  real  plane  containing  the  imaginary  circle,  and  the  line 
of  centers  is  perpendicular  to  it. 

'•\.  When  the  spheres  S  and  N,  are  tangent  at  a  point  .!.  The 
spheres  of  the  pencil  are  all  tangent  at  -I.  The  radical  plane  is  the 
common  tangent  plane  at  .1,  and  the  line  of  centers  is  perpendicular 
to  the  radical  plane  at  .  I. 

The  position  of  the  radical  plane  in  the  second  form  of  the  pencil 
has  been  fixed  oiilv  analvt  icallv,  A  useful  geometrical  proper!  v 
is  that  all  the  tangent  lines  from  a  fixed  point  of  the  radical  plane 
to  the  spheres  of  the  pencil  are  eipial  in  length.  For  if  /'  is 
any  point  of  space,  and  M  the  center  of  a  sphere  of  radius  /•.  the 
square  of  the  tangent  from  /'  to  the  sphere  is  Ml''  /•'.  Applying 
this  to  a  sphere  of  the  pencil  (  ~>  ).  \ve  find  the  square  of  the  length 
of  the  tanent  to  be 


It   the  point    /'  is   in   the  radical  plane  (  ii  ).  this  distance   is   inde- 
pendent  of   X  and   hence  the  theorem. 

It    follows  from  this  that   ///>•/•<><//'<•<///</>>//•/*///<    (m-nx  <>f  tin  rrnt^r* 

of  »i>h,-riX    nrlll'iilmtnl    tu    ,lll    .sy///,/',x     ,,t'  til,     fH'llfU. 

('loseK'  connected  \\ith   this   is   the   theorem:    .1   >•/'/"/'•    "/•///•»/"/<•'/ 

fn   iln/l  t  il'u   x////i  /v-.v    /x   nr'fhnifiiHilf   <"    <iH   Xli/tt'/'i'X   "t    tin     fn  //<•//   il.'i 
I///   tin  i/t    ilml   ln/s   //x   I'l'nti'i'  "ii    tin     I'liilii'/il  jiliiin'   "t    tin-  j»  •//<•//. 


rattle 


268  THRKK-DIMEXSIOXAL  GEOMETRY 

Tin-  lust  part  of  this  theorem  is  a  consequence  of  the  previous 
theorem.  Tlie  first  part  is  a  consequence  of  the  linear  nature  of 
the  condition  (4)  for  orthogonality. 

112.  Bundles  of  spheres.    The  spheres  defined  l>v  the  equation 

,s1+x.s,+  /t.y.s=o,  (i) 

where  N  .  N,,  N   are  three  spheres  not  belonging  to  the  same  pencil 
and  X,  /J.  are  arbitrary  parameters,  form  a  }>nn<ll<'  of  spheres. 
The  centers  of  the,  spheres  of  the  bundle  have  the  coordinates 

.  /;  +  v:,  +  H  *',  t  _  //,  -f  x//,  +_/z//:s  _  //,  -4-  x/,,  + 

</,  +  \<t,+  fJ.'!..  <l    +  \i 


From  (  '2  )  it  follows  that  if  the  centers  of  the  three  spheres  ,S'  , 
Sa,  »s'  lie  on  a  straight  line,  the  centers  of  all  spheres  of  the  bundle 
lie  on  that  line.  The  center  mav  be  anywhere  on  that  line,  and 
the  radius  of  the  sphere  is  then  arbitrary.  Hence  *<  >•/<«•<•/,//  ,v/.\v 

of  il    I'll  //<//>'    of  .sy-//i  /•«  .v    I'ntlxixfit    of   <///    Spht')'t'f>    /r/iom-    n  nt<  •/>•    //c    o/(     (( 

fttmiijhf  htii'. 

More  generally,  if  the  centers  of  ,S'  .  ,s'  ,  and  N  are  not  on  the 
same  straight  line,  they  will  determine  a  plane,  and  the  centers 
of  all  spheres  of  the  bundle  lie  in  this  plane.  This  plane  is  the 
l>I<iii>'  of  fi-nti-rx.  and  any  point  in  it  is  the  center  of  a  plane  of 
the  bundle.  In  this  case  the  three  spheres  .S',,  .S',,  .S',  intersect  in 
two  points  (real,  imaginary,  or  coincident),  and  all  spheres  of  the 
bundle  pass  through  these  points.  If  the  two  points  are  distinct. 
thev  are  symmetrical  with  respect  to  the  plane  of  centers:  if  thcv 
are  coincident,  thev  lie  in  the  plane  of  centers.  Hence  we  sec  that 

a   oiiH'Hi'   of  .vy///»Vi  .s    i'oii>ti>ttn    in    r/i'Hi'fttt   uf  itplnTf*    ll'Jinx,     ci'ttfi't'x    //<'   ill 
it    fl.fi-'  I   /'Itllli'    ilflil    ll'liirjt    jHlsx    f///'"/l'//l    it    fl.fi'i/    I'oint. 

The  radical  planes  of  the  three  .spheres  N,,  .s'.,,  and  X,,  taken  in 
pairs,  are 


,/  N  -  </  .S1  =  0, 

:;      1  13 

das  -  ",N,=  o, 

whicli  evidently  intersect  in  a  straight  line  called  the  /•//<//<•<//  "./-/.v 
(>t  tin'  bundle.  It  is  perjiendieiihir  to  the  plane  of  centers  and  passes 
through  the  jioints  common  to  the  spheres  o|  the  bundle.  The 
radii-al  plane  of  anv  two  spheres  of  the  bundle  passes  t  hi'oii^h  the 
radical  axis. 


THE   SPHERE   IX  ('ARTESIAN   COORDIXATKS        l^iH 

Any  sphere  orthogonal  to  three  spheres  of  a  bundle  is  orthogonal 
to  all  the  spheres  of  the  handle  heeause  of  the  linear  form  of 
condition  (4),  Jj  111.  The  centers  of  sneh  spheres  lie  in  the  radi- 
cal axis  of  the  bundle,  since  bv  ^  111  thev  must  lie  in  the  radical 
plane  ot  any  two  spheres  of  the  bundle,  and  anv  point  of  the  radical 
axis  is  the  center  of  such  a  sphere.  It  is  not  difficult  to  show  that 
these  spheres  form  a  pencil. 

In  fact,  to  mil/  binnUi'  <>f  Rpltm'n  »•>'  >/m>/  <ixx<><-'mti'  <tn  nrt/m</n/ni/ 
jn'in'il  nf  spheres  ninl  tn  «n//  /«•/"•//  "/'  n  Hnlirr?  mi  nrfJint/dnnl  /<?///<//»•. 
7Y/>'  fflntinii  <if  j)f'//<'//  iiinl  fnniiUt'  ix  *;/<•//  tlml  »•/•*•/•//  ,v/<//r/v  «/'  ///,•  in'tn-il 
ix  nrthnijnnill  t"  I'i'i'i'i/  xji/n'/'f  nf  f/n'  hninlli\  tin-  I i in'  •>('  mitiTx  "/'  tin' 

I >t  Hi'il  In  tfi,j  rndlcnl  n.r/'x  nf  ///<•  fmnillf,  <in</  f/if  /•//»//<•<//  i>fi/ni'  nf  the 
yi'iit'il  ix  tfn1  ji/'O/i-  nf  i'1'nfi'rx  <>f  f/n'  fnitHJIr. 

As  far  as  the  details  of  the  above  theorem  have  not  been  ex- 
plicitly proved  in  the  foregoing,  the  proofs  are  easily  supplied  bv 
the  student. 

Closely  connected  with  the  foregoing  theorem  is  the  following: 
AH  ttnha'fft  nrtJini/nnnl  1»  f/m  //'./>•'/  tttt/n'i'fx  1'/>rni  it  bundle  <nnl  <ill 

>>Y(//»7VX     nrtjlni/nllill    in     fj/)'«'l:     jl.l'i'i/    Xji/l I'fi'X  fnflll     it    jn'lli'il. 

The  foregoing  assumes  that  the  three  spheres  ,S'  ,  \.  N,  are 
not  all  planes.  If  thev  are,  the  bundle  of  spheres  reduces  to  a 
bundle  of  planes.  Otherwise  the  bundle  of  spheres  contains  a 
one-dimensional  extent  of  planes  through  the  radical  axis  of 
the  bundle. 

113.  Complexes    of    spheres.     The    spheres    represented    bv    the 

euuat  ion 

,V1+XSs+AU%+J'tf4=0,  (1) 

where  ,V  ,  .S',,  ,S'  .  ,S'  do  not  belong  to  the  same  bundle  or  pencil 
and  X.  //.  v  arc  arbitrary  parameters,  form  a  <•>,////</,./•  ot  spheres. 

'I  he  radical  planes  of  the  tour  spheres  \  ,  N_,  N  .  ,s'  taken  in 
pail's  intersect  in  a  point,  and  the  radical  plane  ot  anv  two  spheres 
ot  the  complex  pass  through  that  point,  llns  point  is  the  /•</«//••<// 
i-i'iifff  of  the  complex.  From  the  properties  of  radical  planes  it 
follows  that  the  sijuare  of  the  length  of  the  tangents  d:a\\n  trom 
the  radical  center  to  all  spheres  of  the  complex  is  constant.  '1  here- 
tore  the  radical  center  is  the  center  ot  a  sphere  orthogonal  to  all 
the  spheres  of  the  complex.  (  'on versd v.  il  is  easv  to  see  that  any 
sphere  orthogonal  to  this  sphere  belongs  to  the  complex.  That.  is. 


270  THKKK-  DIMENSIONAL  (JKOMKTKV 

tlic  mniiili'j'  nntxixtx  <>t"  sjihiTt's  orthnijnnnl  to  <i  //.rr</  fxtxt1  xpnfre  icfioHC 

1-t'ntt'r  /x  thi'  rii'/ii-'i/  <-i'ntfr  <>f  t/ic  f»tnj>!f'j: 

If  the  four  spheres  intersect  in  a  point  that  point  is  the  radical 
center.  The  base  sphere  is  then  a  sphere  of  radius  zero  and  the 
complex  consists  of  spheres  passing  through  a  point. 

The  above  discussion  assumes  that  the  four  spheres  $^,  ,S'o,  N.?,  .S'4 
are  not  planes.  If  they  are,  the  complex  simply  consists  of  all 
planes  in  space.  In  the  general  ease  the  complex  contains  a  doubly 
infinite  set  of  planes  which  pass  through  the  center  of  the  base 
sphere. 

114.  Inversion.  Let  a  be  the  center  of  a  fixed  sphere  .S',  /""  the 
square  of  its  radius,  and  /'  anv  point.  The  point  /'  mav  be  trans- 
formed into  a  point  /''  by  the  condition  that  <>/'/''  forms  a  straight 

line  and  that  ()/>.(>/>'=]?.  (1) 

This  transformation  is  an  itin'r>*i<>)i,  oi'  transformation  by  /v»v'/>- 
roi-itt  rii'lii/x.  The  point  <>  is  the  center  of  inversion,  and  the 
sphere  S  is  the  sphere  with  respect  to  which  the  inversion  takes 
place. 

If  the  point  <>  has  the  coordinates  (./;,,  //  ,  2n),  the  equations  of 
the  transformation  are 

.r'=r  -f /fJ(  •''"-—' 
If' 


where  I!1  =  ( .r  —  rQ )'  +  ( //  —  //„ )'  +  (z  -  21,,)2. 

In  this  transformation  the  constants  mav  be  either  real  or 
imaLriuarv.  If  (./-|t  // .  z  )  is  real  and  /r  real  and  positive,  the 
inversion  is  with  reference  to  a  real  sphere.  If  ( ./•  .  //..  2{~)  is  real 
and  k"  real  and  negative,  the  inversion  is  with  reference  to  a 
sphere  with  real  center  and  pure  imaginary  radius.  In  this  ease, 
however,  real  points  are  transformed  into  real  points. 

From  the  definition  and  equations  ('!)  it  appears  that  anv  point 
/'  has  a  iini'jue  transformed  point  /''.  and.  conversely,  unless  /'  is 
at  the  ori'mi.  or  on  a  minimum  line  through  ' >,  or  at  infinity. 


THK   SIMIKKK    IN   CAKTKSIAN  CO<  >KI>IN  ATKS        'J7I 

To   luilitlk1   these  special   cases  we   take   <>  at   tlir   origin   and    write 
equations  ('2)  with  homogeneous  coordinates  as 


pz  -  k-zt. 

pt'  '  =  /•+  //-  +  r. 

From  ('•})  it  appeal's  that  the  transformed  point  of  n  is  indeter- 
ininate,  luit  that  if  /'approaches  n  alon^  the  line  ./•://:  z  -----  I  :  m  :  n. 
the  point  /'  recedes  to  intinitv  and  is  transformed  into  the  point  at 
infinity  /:///://:<).  Hence  we  mav  sav  that  the  center  of  inver- 
sion is  transformed  into  the  entire  plane  at  intinitv.  ('onverselv. 
anv  point  on  the  plane  at  infinity  but  not  on  the  cin-le  at  infinity 
is  transformed  into  <>. 

If  /'  is  on  a  minimum  line  through  <>  but  not  on  the  imaginary 
circle  at  infinity,  then  ./://':  z'  =--  .r  :  >/  :  ,r  and  t'  =  H.  That  is.  all 
points  on  a  minimum  line  through  <>  is  transformed  into  the  point 
in  which  that  line  meets  the  imaginary  circle  at  infinity,  ('on- 
verselv, if  /'  is  on  the  imaginary  circle  at  infinity  the  transformed 
point  is  indeterminate,  but  ./•'  :  //'  :  z'  =  .r  :  //  :  .~.  so  that  any  point  mi 
the  circle  at  infinity  is  transformed  into  the  minimum  line  through 
that  point  and  the  center  of  inversion. 

Consider  now  a  sphere  S  witli  the  equation 

•i  (  ./•-  +  >r  +  2*  )  +  -.  t'jr  +  -  ////  +  1'  hz  +  ''-<>.  (  4  ) 

It  is  transformed  into 

'ik'  +  -2.  firs  +  -2  .//•-//  4-  -  hk-z  +  <•(  .>•-  -f  //'  4-  r  )  =  <  '.  (  •">  , 

This  is  in  general  a  sphere,  so  that  in  general  spheres  arc 
transformed  into  spheres.  But  exceptions  are  to  be  noted: 

1.  If  '•  0,  </  •  0.  (4)  is  a  sphere  through  n  and  (  .">  )  a  plane 
not  through  <  ),  so  that  sphci'es  tliroilgh  the  center  of  inversion  are 
transformed  into  planes  not  through  the  center  of  inversion. 

'_'.  I  f  1  1  —  I  ','•--  <>,  (  4  )  is  a  plane  not  through  n  and  (  ~>  )  a  sphere 
through  <  >,  so  that  planes  not  through  the  center  of  inversion  are 
transformed  into  spheres  through  the  center  of  m\ersion. 

:>.  If  a  -  0,  ,-~  0,  (  [  )  and  (  ."»  )  represent  the  same  plane  through  ". 
so  that  planes  through  the  center  of  inversion  are  transformed  into 
themselves. 


TIIKF.K    IHMKNSIONAL   < !  KO.M  KTK  Y 

Bv  ;ni  inversion  the  aii'jde  between  two  curves  is  equal  to  the 
alible  between  the  two  transformed  curves;  that  is,  the  trans- 
formation is  ••mifnnnnl.  To  prove  this  we  compute  from  ( ~2)  (with 
j-  =  I),  if  =  0.  .-  =  0  ). 

,  /.,- '  =    >4  ;  ( ,/-  +  z-  -  r )  •  Lr  -  -2  .>•>/  < ///  -  '1  j-z  •  h } , 
I! 

< /  y '  =    (4  I  -  -  -'•// ' /./•  +  ( .'•-  -  //-  +  z' ) '  1>i  -  -  // ? ' h\ - ,  ( » ) ) 

/I 


Hence,  if  we  [dace  <h'~-=  </.r"J4-  '/'/""+  '^'"  :'"d  </x'-  —  </./•"  +  (///"4-  '^", 
we  have 


Now.  if  '/./•.  '///.  '/r  corresponil  to  displacements  on  a  curve  from 
/',  and  8.r.  8//.  ^  to  displaeements  alon^  another  curve  from  /',  the 
an«_rle  n  between  the  curves  is  ^iven  bv 


Similarl,  the  anle  n1  between  the  transformed  curves  is 


and  it  is  easy  to  prove  from  (<i)  that  cos  n  =  cos  n'. 

Anv  pencil,  bundle,  or  complex  of  spheres  is  transformed  into  a 
pencil,  bundle,  or  complex,  respectively.  'I  he  line  ol  centers  of  the 
pencil  is  not,  however,  in  general  transformed  into  the  line  of  cen- 
ters of  the  transformed  pencil,  but  becomes  a  circle  cutting  the 
spheres  df  the  transformed  pencil  orthogonally.  Also  the  radical 
plane  (it  the  pencil  is  not  transformed  into  the  radical  plane  of  the 
transformed  pencil,  but  into  one  of  the  spheres  of  that  pencil. 

Similarly,  the  plane  of  centers  of  a  bundle  is  transformed  into  a 
sphere  cutting  all  the  spheres  of  the  bundle  orthogonally,  and  the 
radical  axis  of  the  bundle  is  transformed  into  a  circle  orthogonal 
t»  the  transformed  bundle. 

<  )n  the  other  hand,  the  base  sphere  ol  a  complex  is  transformed 
into  the  base  sphere  o)  the  transformed  complex. 


TlIK   SIMIKKK    IN    CAKTKSIAN    (  <  >( >K  I  >I  N  A  TKS 

If  we  take  a  jit'iicil  of  spheres  interseetiiiLj  in  a  real  circle  and 
take  the  center  df  inversion  on  that  circle,  the  pencil  of  spheres  is 
evidently  transformed  into  a  pencil  ot  planes.  It  we  take  a  bundle 
of  split-res  intersect in>_r  in  two  real  points  .1  and  //.  and  take  .1  as 
the  center  of  inversion,  the  bundle  of  spheres  becomes  a  bundle  of 
planes  through  the  inverse  of  /,'.  It  we  take  a  complex  of  spheres 
and  place  the  center  ot  inversion  on  the  base  sphere,  the  complex 
becomes  one  with  its  base  sphere  a  plane;  that  is,  it  consists  of  all 
spheres  whose  centers  are  on  a  tixed  plane. 

EXERCISES 

1.  Trove  that;  by  an  inversion  with  respect  to  a  sphere  N  all  spheres 
which  pass  through  a  point  and  its  inverse  are  orthogonal  to  >'. 

2.  Prove  that   a   point   and   its   inverse  are  harmonic  conjugates  with 
respect    to  the   points   in  which   the  line  connecting  the  tirst   two  points 
intersects  the  sphere  of  inversion. 

3.  Prove  that  the  inverse  of  a  circle  is  in  general  a  circle  and  note 
t  lie  special  cases. 

4.  Prove  that    if  two  figures  are   inverse  with   respect   to  a   sphere  >'  . 
their  inverses  with   respect   to  a  sphere  N,  whose  center  is  not  on  >'    are 
inverse  with  respect   to  V.  the  inverse  of  >'    with  respect  to  >',. 

5.  Prove  1  hat  i  f  t  wo  t  inures  are  inverse  with  respect  to  a  sphere  >'  .  t  heir 
inverse  with  respect  to  a  sphere  >',  whose  center  is  on   ^  are  svmniet  rical 
with   respect  to  the  plane  /''.  the  inverse  of  >'    with  respect   to  >'.,.    Coii- 
verselv,  if  two  figures  are  symmetrical  with  respect  to  a  plane  /'  t  he  v  are 
inverse  wilh  respect   to  anv  sphere   into  \\hich   the   plane  /'  is  inverted. 
Therefore  inversion  on   a  plane    is  detined  a>  reflection  on  that   plane. 

(i.  I'ro\-c  that  it'  X  i-;  a  sphere  of  radius  /•  and  >''  is  its  inverse,  the 
radius  of  >'  is  cipial  to  the  radius  of  >'  multiplied  hv  the  sijiiare  of  the 
radius  nf  the  sphere  of  inversion  and  divided  hv  the  absolute  value  ot 
the  power  of  the  center  of  inversion  With  respect  to  >'. 

7.  Prove  that  anv  two  nonintersect  hit:  spheres  ma\  be  inverted  1>\ 
an  inversion  on  a  real  sphere  into  concentric  spheres. 

s.  Prove  that  any  three  spheres  may  be  inverted  into  three  spheres 
ot  ei|'ial  radius. 

'.i.  Prove  that  inversion  on  a  sphere  with  real  center  and  pure  imauri- 
na  r\  radius  /•/  is  equivalent  to  inversion  on  a  sphei'e  with  the  >ame  center 
and  real  radius  /•,  folloued  l>v  a  transformation  \>\  \\hich  each  point  is 
replaced  liy  its  syillinet  rieal  point  u  it  h  respect  to  t  he  center  of  inversion. 


'2~  1  THKKK    IHMKNSIONAI.   <  I  K<»M  KTK  Y 

10.  A  Mirface  which  is  its  own  inverse  is   called  nn'i//iii/ninfif.    Trove 
that    aiiv    anallax'inat  ic    surface    nits    the    sphere    of    inversion   at    ri<,rht 
angles  it'  tin-  point   of  intersect  ion  is  not  a  singular  point  of  the  surface 
ami   is  the   envelope   of  a    fainilv    of  spheres   which   cuts   the   sphere  of 
in  versi<  in  ort  ho^miall  \  . 

11.  I'ruve    that    the    product    of  two   inversions    is  equivalent    to  the 
product  of  an  inversion  and  a  metrical  transformation  or  in  special  eases 
In  a  metrical  transformation  alone. 

115.  Dupin's  cyclide.  The  transformation  by  inversion  is  useful 
in  studying  the  class  of  surfaces  known  as  flu/iin's  ry/r//,/,-*.  These 
are  defined  as  the  envelope  of  a  family  of  spheres  which  are  tangent 
tot  hive  tixed  spheres. 

It  the  centers  of  the  tixed  spheres  do  not  lie  in  a  straight  line  we 
mav  l>v  inversion  bring  them  into  a  straight  line.  To  do  this  we 
have  simplv  to  draw,  in  the  plane  <>|  the  centers  of  the  three 
spheres,  a  circle  orthogonal  to  the  three  spheres  and  take  anv  point 
on  that  circle  as  the  center  of  inversion.  The  circle  then  goes  into 
a  straight  Hue  which  is  orthogonal  to  the  three  transformed  spheres 
and  hence  passes  through  their  centers.  This  is  a  consequence  of 
the  conforma!  nature  of  inversion.  For  the  same,  reason  the  surface 
enveloped  l>v  spheres  tangent  to  the  original  three  spheres  is  in- 
verted into  a  surface  enveloped  by  spheres  tangent  to  three  spheres 
whose  centers  lie  on  a  straight  line. 

\Ye  shall  study  first  the  properties  of  such  a  surface  and 
then  liv  inversion  deduce  the  properties  of  the  general  Ihipin's 
cvelide. 

Let  us  take  the  line  of  centers  of  three  fixed  spheres  as  the  axis 
ot  z  and  the  equations  ot  the  spheres  as 


-  -   (  /  —  '•  '•'  -     *  — 


T1IK   Sl'llKliK    IN'   CAUTKSIAN    (  '()(  MMUNATKS 

to  the  sum  or  tin-  difference  of  the  radii  of  tin-  two  >phere>.     Thi 
^ives  the  three  euations 


n-  +  /•-  +  '••     -  ••/•  +  '7  =  (/i  r.)". 

which  have  in  general  four  sohilions  of  the  form 

<•_=  const.,      /•  ---  const..      ir-f-A       const.  (}, 

'1  heretore  the  sphere  ('2)  helon^s  to  one  o|  Imir  laiuilies  each 
ot  which  consists  ot  spheres  \\iih  ;t  constant  radius  and  with 
their  centers  on  a  fixed  circle.  Kaeh  family  ohviouslv  envelops  a 
rhiLj  surface. 

I  here  are  therefore  in  u'eneral  tour  Ihipin's  evelides  determined 
hv  the  condition  that  ihe  enveloping  spin-res  arc  lan^'ciit  to  three 
lixed  spheres. 

Let  us  take  anv  one  of  the  solutions  (4)  and  change  the  coi'inli- 
nate  system  so  that  <•  =  U.  The  equation  of  the  family  of  spheres 
may  then  he  written 

(  ./•  —  </   ci  is  0  >'  +  (  //  —  "n  sin  0  )"+  2"=  /•",  (  •")  ) 

where  61  is  an  arhitniry  parameter  and  </    and  /•  are  coi^tants. 
The  surface  cii\'eloped  l>y  (^)  is 

(  j  •"  +  tj~  +  z~  +  (/,';'  -  /•"  »'"'      4  n  -  (  .r'  +  //"  )•  (  '  '  ) 

This  is  the  equation  of  the  /•/////  an  !•/<(<•>•  formetl   li\    re\'ol\iiiL,r  aliout 

the  axis  i  if  r  the  circle 

(  ./'  —  <i  :  )   -f-  .;•-/'.  (  i  ) 

lleliee   ''////    Ihijiiiix  >-//<'/i</f   t#  ///-    ////•,  />•»•  >,t   tin    rtn<j  ,v///-/./.-»  -y.  •/•///•  •/ 

Li/   i-<i',,lrtii'/  'I    I't/'t'lf    ill*,,  nf   <ni    il.l/'x   ii'it    in    it  a  filil/ii'. 

'I'he  riii'4'  >urfacc  contains  t\\o  families  of  circles  forming  an 
orthogonal  network.  The  one  famiU'  consists  of  the  meridian  cir- 
cles cut  out  hy  planes  through  the  a\i<  of  reyohltioil,  the  other  ol 
circles  of  latitude  made  l>v  sections  perpendicular  to  that  axis. 

Since,  hy  inversion,  circle-  are  transformed  into  circles,  and  angles 
are  conscr\'ed.  there  cxiM  on  any  Piipuis  c\clide  t  \\  o  similar 
families  of  circles  also  forming  an  orthopmal  network. 

The  rin  LT  surface  i-  the  envelope  not  onl\  of  the  family  o!  spheres 
whose  ei  pi  at  ion  i->  (  ."»  j  hut  also  of  i  he  famih  \\  it  h  the  equal  ion 

./••  f    i/-  '   (  :       ./    Ian  0)~-     (  -'    see  0       r  i".  (  ^  ) 


l27l»  TIIKKK   DI.MKNSIONAL  (IKdMKTKV 

This  family  consists  of  spheres  with  their  centers  on  <>Z  each  of 
which  mav  be  generated  by  revolving  about  <>/.  a  circle  with  its 
center  on  <>/  and  tangent  to  the  circle  (7).  The  spheres  of  this 
family  are  tangent  to  the  rin'_r  surface  alon^  the  circles  of  latitude, 
while  the  spheres  of  the  family  (  .">  )  are  tangent  to  the  rim_r  surface 
aloii'_,r  the  meridian  circles.  The  family  of  spheres  (  s  )  mav  be  deter- 
mined bv  the  condition  that  they  are  tangent  in  a  definite  manner 
to  three  spheres  of  (  •">  ). 

Ilelice  <ln]j  Ihi/iinx  fj/rli<1t'  until  f'f  iffttft'ittt'il  i/i  ( I''"  H.'ili/x  <ix  (/if 
tni'tl"[K  nt  il  t't in thl  nt  xj'tii  r,  x  ruiixixt  tiiij  <>>  xjiJni'i  x  tit it'i*  nt  t»  thl'fi' 
fl.i'i'l  xjifii'i'ix.  Eiti'lt  JttiHtltj  '[1  Xjilii'i't'x  ix  fii/ii/f/if  fn  tin'  f//i'!l<lf  <ll"it</ 
<i  tit  mil  if  'if  fi/'i-li  x,  tin-  tii'"  fnmtlifx  «f  <•//•<•/,  *  hftn<f  nrtftni/'UKtl. 

The  planes  of  each  family  ot  circles  interseet  in  a  straight  line. 
This  follows  from  the  theorems  ot  x  1  1  '2.  since  the  inverse  spheres 
of  the  spheres  (  ;> )  belong  to  the  same  bundle  and  t  he  circles  arc  inter- 
sections of  spheres  of  that  bundle,  so  that  their  planes  pass  through 
the  radical  axis  of  the  bundle.  Similarly  for  the  sphere-  (  S  >. 

The  circle  (7)  intersects  the  axis  of  r  in  two  real,  imaginary,  or 
coincident  points.  Theid'oiv  a  I  Mipin's  evdide  has  at  least  this 
number  of  singular  points.  \Ve  shall  see  later  that  it  al-o  lias 
other  singular  points,  but  we  shall  confine  our  attention  at  present 
to  these  two.  Call  them  A  and  /•'.  The  spheres  of  one  of  the  fami- 
lies which  envelop  tin-  evdide  intersect  in  A  and  /.',  as  is  seen  in 
the  ease  of  the  rin^  surtace.  Consequently,  if  one  of  the-e  points, 
as  A.  is  taken  as  the  center  of  inversion  this  family  of  spheres 
become-  a  family  of  planes,  and  the  evdide  invert-  into  a  surface 
enveloped  by  spheres  \\hidi  are  tangent  to  three  of  these  planes. 

If  A  and  //  are  distinct  the  planes  pa--  through  the  puint  />', 
the  inverse  of  //,  ami  the  evdide  is  inverted  into  a  cone  of  revolu- 
tion, which  is  real  it  A  and  />'  are  real,  and  imaginary  if  .1  and  11 
are  imaginary. 

It  A  coincide-  with  /<'  the  plane-  are  parallel  and  the  cvdide  is 
inverted  into  a  cylinder  of  revolution.  \\Y  have  accordingly  the 
theorem:  .1  Itufiin  *,•//••//,/,'  ///<///  «////•</_//.<  /«,•  /'///•,/•/,,///(,'.<  ,i  !•.,//,•  <>t  /•-/•"- 

( 'oii.-ei | uent  1\'  \\'e  1 1 1 a v  obtain  a ii \"  I'Vclidc  in  \\liieh  the  singular 
point-  A  and  //  are  distim-t  b\-  in\  ert  iic_r  the  cone 

o  <  '»  ) 


THE   SPHERE    IN   CARTESIAN  COORDINATES         'J77 

from  anv  real  or  imaginary  center  of  inversion  witli  respect  to  anv 
real  or  imaginary  sphere;  or,  what  amounts  to  the  same  tiling,  we 
may  transform  tlie  origin  to  anv  real  or  imaginary  point  and  invert 
Irom  the  origin.  The  equation  of  the  cone  is  then 

( ./•       a  )'  +  (  //  -  tf  y  -  m-  ( .;-     -  7  r  -  0,  (111) 

and  its  inverse  with  respect  to  the  origin  is 

( ic  -f-  J-      ,/i-y- )  ( ./•-  +  i/-  -f-  r  y  -  'I  k-  (  a.r  -f  tf;i  -  -  uc-jz  )  (  JL:-  +  //-  -f  z~ ) 

+  //(  r- +//--///  -z-)-  :  0.  (11  ) 

To  consider  the  case  in  which  the  points  A  and  /.'  coincide,  we 
invert  the  cylinder 

and  obtain   for  its  inverse 

(«-'+  f  -  r  )  ( .r  +  //'-  +  2- )--  '1  k~  (aj'  +  tfii )  ( .'•-  +  //-  -f  2'-) 

The  cvclide  is  therefore,  a  surface  of  the  fourth  order  unless  the 
lirst  coel'ticient  in  either  (  11)  or  (  1  '1 )  \anishcs.  I»ut  this  happens 
when  and  onlv  \\hen  the  cone  (1")  or  the  evlinder  (111)  passes 
through  the  center  of  inversion. 

It  now  we  make  the  equations  (11)  and  (!•'))  homogeneous, 
and  place  /  —  (I  to  determine  the  section  with  the  plane  at  inlinitv, 
we  j^et  the  circle  at  inlinitv  as  a  double  curve  when  the  surface  is 
of  fourth  order,  and  the  circle  at  inlinitv,  together  with  a  straight 
line,  when  the  surface  is  of  the  third  order. 

Hence  <i  Ditfiiit's  cii<-li<l<'  ix  <i  xiirfiii'f  <>f  ///»•  fourth  «r<l<'r  tntrt 
thi-  i-i i-cl<'  at  iiifiniti/  <tx  it  tlnitl'li'  r/trrc,  «//•  it  xt<rt'it<;-  <>f  th<  t/tir</  "/•</<  r 
ir/th  'In'  <•//•<•/,•  (//  i /it!  nit  i/  <tx  (/  xi  in  fli'  <•/! /•/•>'. 

\\'e  proceeil  to  lind  the  singular  points  of  expiation  (11).  \Ve 
can  \\ithoiit  loss  of  ^eiieralitv  so  turn  the  axes  that  ^  —  0.  and 
will  make  the  abbreviations 

.1  .=    ,i-  -  nri\ 
/.'      -'•-  -f  //-  f  r, 

/  ~r\      -- 

and   write  the  equation   a-- 

.I/,'-       'Ik' 1. 1!   f  //(/•+  V"       //'"':•"')       ".  (  11  ) 


278  THKEE-  DIMENSIONAL  CJEOMETKV 

Tin-  singular  points  are  then  the  solutions  of  this  equation  and 
the  following,  formed  hv  taking  the  partial  derivatives  with  respect 
to  .r,  //.  and  z  : 

4  . 1  /,'./•  -  -  k-a  R        -  4  trLjr  +  '2  k\r      =  <>, 

4  . /  ////  -  4  /-•  Li/  +  -2  /-'//      -  0,  (15) 

4  JA'*  +  '2  k-in-ylt'  -  4  k-Lz  -  -2  k*,,rz  =  0. 

liv  multiplving  equations  (!•>)  in  order  bv  .r,  i/,  z  and  adding,  and 
subtracting  the  result  from  twice  (14),  we  obtain 


(.1A'-A--A  )/«'  =  0. 

Also,  by  combining  the  tirst  two  of  (15)  we  have 

'2k-uif/i'  =  0.  (17) 

From    (17)    we    have   either  A'  —  0   or  // =  0.    Taking   lirst   the 
condition  //  —  <>,  but   //  ^  (.),  from  (10)  and  (15;, 


whence 

The  point  (    .,    '     ;,'  (I,     _         j  is  therefore  a  singular  point.    It  is 

the  inverse  of  the  vertex  ot  the  cone  and  is  the  point  II  of  the 
discussion  on  page  270. 

Consider  now  the  solution  A' =  0  of  equation  (17).    From  (15) 

we   have  cither  ./•  -    <>,   //  =  I),  z  —  0,  or  L  =   '   •  z  —  n.     The  origin  is 

therefore  a  singular  point,  the  inverse  of  the  section  of  the  cone 
with  the  plane  at  intinitv,  and  is  the  point  .1  of  the  discussion  on 
page  •J"'!. 

The  alternative  A'  —  O,  L=  '  '2=0  leads  to  the  two  singular  points 


•  ±   }     •  'I  j.    'I  hese  points  tail  to  ex  ist   it  /<     :  n.  but  in  that  case 

the    inversion    is    from    a   point    on    the    axis    of    the   cone,    and    the 
surface   (  11  )   is   thru   a    rin^   surface. 

The   two  sium ilar  points  ju.-t   found  are  each  connected   with   ./ 

]      ;  i    \  ]  ' 

and   /'   t)v  minimum  lines. 

It     \\  e    consider    in     the    same     wav     eijuation    (  1  :'>  ),    \\  c    obtain 
simihii-  n-siilt-  except   that  the  singular  point   /,'  coincides  \\itli  .1  at 


T1IK  Sl'lIKl;i:  IN  <  AKTKS1AN  (  '(  )<  >K  I  >  I  N  ATKS  -J7'.l 
the  origin,  since  tlic  assumption  _//  —  0  leads  to  the  conclusion  /,'  ". 
The  t\vo  points  i  *±  »  0  I  aiv  a^aiii  sinjjulur  points  unlrss  «  u, 


when  the  surface  (  1  •'!  )  is  a  rin;_,r  surface  with  a  single  singular  [mint. 

.1  Ihijiiiix  ci/flitlt'  iilin-h  i*  H"t  it  r///'/  xit/'tiii-i'  liiix  in  i/i'/it  rul  f"ii  r 
fi/i/ti-  xtni/i(h(f  fn'tntx  tn'"  "j  //'Iti'-h  iii'i1  t'litt/u'cti'il  with  tin  "///-/•  ///••/ 
In/  in/  ntin  n  ill  ll/tt'X.  I  it'<>  "I  f/ii'Kf'  Xtiujniiir  ji'n/itu  nni/l  fuiiiftilt;  lit 
t('/i/r/t  film-  tin'  (•//»•//(/('  Inix  th/'fi'  linttf  xuti/nlili'  /'"intx  tir<>  -//'  ti'Jitfh 
i.tt't'  i'"iiii'  '•(<  <l  K'tfft  tin'  tin  r<l  !>i/  ml  ii  t  inn  111  liitrx, 

It  tOllous,  tit'  course,  that  the  Dujiin's  c\cliilcs  are  not  the  gen- 
eral surfaces  of  fourth  order  with  the  circle  at  inlinitv  as  a  doiihlc 
eiiiA'e  nor  the  general  surtace  ol  third  order  through  the  circle  at 
intinitv.  '1'lioe  more  general  surfaces  will  he  noticed  in  the  next 
sect  h  in. 

EXERCISES 

1.  Trove  that  anv    l)ujiin's  evelidc   is  iintillagiuatie  with    respect   to 

each    >phere   of  two    pencils   of  Spliel'eS. 

2.  I'rox'e  that   the  centers  of  cadi   I'aiidlv  of  ellVelojiillg  sjiheres  of  a 
J  )ii]  iin's  c\  'el  ide  lie  on  a  ei  Mile. 

Ii.  Prove  that  the  two  lines  in  which  the  planes  of  the  two  families 
of  circles  on  the  iMipin's  cvclide  intersect  are  orthogonal. 

4.  I'rove  that  the  circles  on  a  I)upin*s  i-velide  are  lines  of  curvature. 
(  A  line  of  curvature  on  a  surface  is  Mich  that  t  wo  iiorinaN  to  the  surface 
at  two  consecutive  point^  of  the  line  of  curvature  intersect.) 

f>.  I'rove  that  the  oiih'  surfaces  which  have  two  families  of  circles 
for  1  1  nes  of  curvature  are  I  hi  pin's  cv  el  ides.  (  ICxcepl  ion  should  1  >e  made 
of  the  sphere,  plane,  and  minimum  deVelopahle,  tor  \\hi.-h  all  lines  arc 
lines  of  eiirvat  lire.  ) 

116.  Cyclides.     A   eyelide  is  detined  h\-  the  equation 

"„(  •>'-+  '/'+  .•-)-+  ",(•'•"  4-  //'  f-r)-r-".,      (l-  (  1  ' 

\\here  ttit  is  a  constant,  n  a  |'ol\-noinial  of  the  lir-t  decree,  and  //  .  a 
polviiollliill  of  the  second  decree  ill  ./'.  //.  :.  The  l)llpllis  cyclides 
arc  special  cases  of  the  general  eyelide. 

If    n     •    (i   in   eipiaiion    (1  )    the    surface    is   of    the    fourth    derive 
and    rcpi'cseiits    a    1  >ii  piad  rat  ic   surlace   \\ith   the    iiiia'_;'inar\    circle    at 
V  as   a  d<  mlilc   ciir\  e. 


280  THREE-DIMENSIONAL  GEOMETRY 

If  n=  0,  equation  (1 )  is  a  general  of  the  tliinl  degree  and  repre- 
sents a  cubic  surface  passing  through  the  imaginary  circle  at  infinity. 

I  Vgenerate  cases  of  t  lie  cydides  may  also  occur  if,  in  equation  (  1  ), 
f/  -=  0  and  //  is  identically  y.ero.  The  equation  then  represents  a 
qnadric  surface  or  even  a  plane.  These  cases  are  important  only 
as  they  arise  by  inversion  from  the  general  cases. 

In  order  to  study  the  effect  of  inversion  on  the  cyclide  \ve  may 
take  the  center  of  inversion  at  the  origin,  since  the  form  of  equation 
( 1  )  is  unaltered  by  transformation  of  coordinates.  Such  an  inver- 
sion produces  an  equation  of  the  same  form,  which  is  of  the  fourth 
degree  if  //..contains  an  absolute  term  and  of  the  third  degree  if  u., 
does  not  contain  the  absolute  term  but  does  contain  linear  terms. 
In  the  former  case  the  origin  is  not  on  the  surface;  in  the  latter 
case  the  origin  is  on  the  surface,  but  is  not  a  singular  point.  Hence 

The  ittt't'rxc  «f  an i/  I'urliJe  from  a  jK>i /tt  nut  »n  it  /x  <iltt'iit//x  <i  cui-lidi' 
i if  tin'  fourth  order.  Tin1  infers?  of  ant/  cyi-lid?  from  <i  jiotnf  on  it 
which  tx  not  </  sinijuhir  point  in  ulwtfj/x  <t  cyclide  <</  t/n1  t/t/nl  "/'<Ar. 

In  general  the  cyclide  will  not,  have  a  singular  point.  If  it  does 
we  may  take  it  as  the  origin.  Then  in  equation  (1)  the  absolute 
term  and  the  terms  of  tirst  order  in  ?/  disappear.  By  inversion  from 
the  origin  there  will  then  be  no  terms  of  the  fourth  or  the  third 
degree.  Hence  the  ry<-l!<ji-  with  <i  tiinyiihir  ])"!nt  /*  tin-  inrcrxe  of  a 
ijiKidric  xiirfiti-t'.  ( 'onverselv,  as  is  easily  seen,  flu-  //^v/v-v  of  n  ^iiitd  r'n' 
sttrtiii'i'  IK  d  cyclide  trtt//  at  ledxt  one  ximjular  point. 

('oiisider  now  a  cyclide  with  two  singular  points  .1  and  /•'  which 
do  not  lie  on  the  same  minimum  line.  If  we  invert  from  ./  the 
evdide  becomes  a  quadric  surface  with  a  singular  point  at  />'',  the 
inverse  of  /.'.  It  is  therefore  a  cone.  Hence  tin-  <•//<•!/'</, •  u'if/i  ti/'o 
KlUf/nlitf  [lomtx  not  on  f/o'  ^itiin1  minimum  line  ix  t/if  tni'i'/'ti'  <>f  <t  ijHiliinc 
i-o/n-.  ('onverselv,  ttn-  ////v/'.sv  of  n  umiilric.  dim1  from  o-  fioint  not  on  tt 
tf<  <t  i-yrlidf  with  nt  lenxt  tn'o  nini/ular  j>ointx. 

\\ c  have  shown  in  ^  11.")  that  a  Dnpin's  cyclide  of  the  fourth 
onler  hits  in  general  four  singular  points.  We  shall  now  prove, 
conversely,  that  <i  i-i/<-/i</,-  of'  tin-  fourth  order  iritli  four  xin</n/<<r 
[ioi nt*  ix  ((  Itujiin'x  ri/i'liile. 

It  the  four  points  are  ./,  /.',  (\  I>  they  cannot  all  be  connected 
by  minimum  lines,  since  that  is  an  impossible  configuration.  \\V 


THE  SPHERE    IN  CARTESIAN  COORDINATES        2S1 

will  assume  that  ./  and  /.'  are  not  on  a  minimum  line,  and  will 
invert  from  ./,  thus  obtaining  a  quadric  cone  /•'  with  its  vertex  at 
//',  the  inverse  of  /.'.  Anv  plane  section  of  the  evelide  through  All 
is  a  curve  of  the  fourth  order  with  two  singular  points  at  A  and  /-' 
and  two  other  singular  points  on  the  circle  at  inlinitv.  It  therefore 
breaks  up  into  two  cirdc.s  and  is  inverted  into  two  straight-line 
generators  of  the  cone  /•'.  The  cone  is  enveloped  bv  a  one-parameter 
family  of  planes  tangent  along  the  generators.  Then-fore  the 
evdide  is  enveloped  bv  a  one-parameter  familv  of  spheres  tangent 
along  the  circular  sections  through  .1  and  /.'. 

The  plane  sect  ion  determined  bv  the  points  .1.  /.'.and  C  has  t  hree 
singular  points  besides  the  two  on  the  ciidc  at  inlinitv.  Therefore 
it  consists  of  a  circle  and  two  minimum  lines,  and  since  All  is  not 
a  minimum  line,  A<"  and  /!('  are.  Ilv  a  similar  argument  A/>  and 
/>'/>  arc  minimum  lines.  Hence  ('/>  is  not  a  minimum  line. 

\Ve  mav  accordingly  invert  the  cvdidc  from  ( '  and  obtain  another 
cone  with  the  properties  of  /•'.  In  particular,  the  straight-line  gen- 
erators of  this  cone  are  the  inverses  of  circles  on  the  evdide,  and 
its  tangent  planes  are  the  inverses  of  spheres  tangent  to  the  evdide. 
Therefore  the  cone  /•'  is  enveloped  bv  spheres,  the  inverse  with 
respect  to  .1  of  the  last-named  familv.  Then-fore  /•'  is  a  cone  of 
revolution  and,  bv  £  11;"),  the  theorem  is  proved. 

EXERCISES 

1.  I'rove  that   the  envelope  of  spheres  whose  centers  lie  011  a  i[Uadrie 
surface  and  which  are  orthogonal  to  a  ^iven  sphere  is  a  e\  elide. 

2.  iMseiiss  the  plane  curves  called  lii<'in'ti!<i r  <//f /•//'•.-•,  defined  bv  the 

ei  |  liat  H  ill 

and  t  race  the  analogies  to  1  he  ey  elides. 

15.  I'rove  that  the  envelope  of  a  circle  which  moves  in  a  plane  so  that 
i's  center  trace-,  a  ti\ed  conic,  while  the  circle  is  orthogonal  to  a  tixed 
circle,  is  a  bicirciila  r  i|iiart  ic. 

•1.  The  intersection  of  a  sphere  and  a  quadric  snriace  i^  a  .^/J,'/'"- 
ifiniil ,-'n\  I'rove  that  a  spheroqtiadric  inav  be  inverted  into  a  bicirciilar 
qua  rt  ie  and  c<  in  verselv. 

f>.  I'rove  that,  the  intersection  of  a  evelide  and  a  sphere  ;->  a  sphero- 
niiad  rie. 


CHAPTER   XVI 

PENTASPHERICAL  COORDINATES 

117.  Specialized  coordinates.  lYnUsphrrir.il  coordinates  arc  based 
upon  tivr  spheres  of  reference,  as  the  nainr  iniplirs.  It  is  customary 
to  ilrt'mr  thrin  by  usr  ot  thr  ('artrsiaii  equal  ions  of  thr  livr  spheres, 
l»ui  \vr  prefer  t<>  build  up  the  coordinate  svstnn  iiide[)eii(U'iitlv  of 
the  ('artrsiaii  svstnn,  usinin'  onlv  rlrinriitarv  ideas  of  inrasurrinriit 
of  real  distance.  This  brings  into  emphasis  the  fact  that  penta- 
spherical  coordinates  are  not  dependent  upon  ('artesian  coordinates, 
but  that  the  two  svsteins  stand  side  bv  side,  eadi  on  its  own  founda- 
tion. One  result  is  that  certain  ideal  elements  pertaining  to  the 
so-called  imaginary  circle  at  intinily  which  are  found  convenient  in 
('artesian  geometry  arc  nonexistent  in  pentaspherical  '_;vomrtrv  : 
and.  conversely,  certain  ideal  elements  of  pentiispherieal  L;'romrtrv 
do  not  appear  in  ('artesian  ^vonietrv. 

Let  <>.\'.  <>  y,  and  <>/  be  three  nmtiiallv  perpendicular  axes  of 
reference  intersecting  at  <  >.  I'  any  real  point.  <>/'  the  distance  from  <  > 
to  /'.  and  <>L.  i>M.  <>.\  the  three  projections  of  <>/•  ,,n  />.\\  <  >}\  <  >/ 
I'rspeet  ivelv.  A  l'_;vbraic  si^'iis  are  to  be  attached  to  the  three  projec- 
tions in  the  usual  way.  but  <>/'  is  essentially  positive.  We  mav  then 
take  as  coordinates  ot  /'  the  tour  ratios  defined  bv  the  equations 

^:^:^:^:^=tfJ'':OL:(tM:0.\-:  \  (I) 


It  is  obvious  that  to  anv  real  point  corresponds  a  set  of  real 
roi'irdinatrs  and  that  to  anv  set  ot  real  coordinates  corresponds 
our  i-ral  point.  1  In1  extension  IM  mia'_Miiarv  and  infinite  points  is 
made  in  the  n^nal  manner.  In  particular,  as  /'recedes  from  "  indefi- 
nitely in  anv  dirrrt  ion.  the  coordinates  approach  the  limiting  ratios 
1  :  O  :  0  :  <)  :  n.  wliicli  arr  the  coi'ird  mates  ot  a  real  point  at  infinity. 
This,  however,  i-  not  the  onlv  point  at  infinity,  as  \vill  appear  when 
\yi-  consider  thr  tormnla  tor  tin-  distance  betwrrii  t\\'o  points. 


PKNTASPHKKK'AL  ('()(  JKDINATKS  IN; 

The  relation  (  1  )  may  IK-  reduced  to  a  sum  of  squares  by  replacin 
the  coordinates  f.  by   new  coordinates  .r.,  where 


whence 


pr,     :  l!  f,    -:  rr(  J 
p.r,  -    -  £.       (T(  '2 


In  these  coordinates,  \vhieh  \\'e  shall  use  hriiect'ortli.  a  real  point 
lias  four  of  its  coordinates  real  and  the  fifth  pure  imaginary  (the 
proportionality  factor  p  being  assumed  real).  '1  his  slight  nieon- 
\'en  leiice,  if  it  is  an  inconvenience,  is  more  than  balanced  l>y  the 
symmetry  ot  eijiiatioii  (•>).  The  eoi'irdmates  of  the  real  point  at 
intinit  \"  are  no\y  1  :  (I  :  (I  :  0  :  /. 

It  /,'  and  /  '.  are  two  real  [mints  with  coordinates  //.  and  j\  respee- 
tivelv.  the  project  ions  of  the  line  /.'/!  mi  o.\.  <>}.  i  >/,  respect  iyely, 
are  easily  seen  to  be 


and  hence,  since  the  square  ot  the  distance  ot  the  line  /.'/!  is  equal 
to  the  sum  o|  the  sq  uares  ot  it  s  project  ions,  \\  c  com  put  e  reai  I  il\  .  \\  it  h 
the  aid  of  (•"">),  the  «//.s7- m<->  t'"/-i//n/<t  for  the  distance  -/  lietween  two 


•JS4  TH  K  KK-I  >I  M  KNSION  A  L  (I  K<  ).M  HTK  V 

Tin1  formula  (•>),  thus  derived  lor  real  points.  will  IK-  taken  as 
the  definition  of  distance  between  all  kinds  of  points.  From  this  it 
appears  that  </  is  infinite  when  and  onlv  when  one  of  the  points 
satisfies  the  equations  ./^  +  i-'\—  "  an(l  w(-''.  //)-'-  "•  Hence  tin'  Inrttx 
<>f  paint*  >if  infiniti/  ix  ;/t>'>'/i  Lif  t/n'  i'<f>ifit/«/i  ./•  -)-/./•=  0. 

Since  tlie  coordinates  of  all  points  satisfy  (•">),  we  have  for  points 
at  infinity  .i\  +''•''.  —  0  and  j\;  -+-  .r.j-  +  -/'4J  —  "•  Therefore  the  point 
1:0:0:0;  /  is  the  only  real  point  at  intinitv.  The  nature  of  the 
imai^inarv  locus  at  intinitv  will  appear  later. 

118.  The  sphere.  A  sphere  is  defined  as  usual  as  the  locus  of 
points  equally  distant  from  a  fixed  point.  This  definition  includes 
all  spheres  in  the  usual  sense  and  all  loci  which  are  expressed  l>v 
equation  (  *5  ),  £  1  1  7.  in  which  //.  is  fixed  and  <l  =  r  a  constant.  This 

equation  is 

[2  //j  +  (  //,  -f  //O  r]  .r,  +  -1  //.,.>•.,  +  -2  //,/-,+  -2  y(.r4 

+  ['2  //.  +  /  (;/,  +  '//,)  r]  ./;.  =  o.  (1  ) 

This  is  of  the  tvpe 


\\-here  P''{=  -  //,  +  (  '^  +  ''//•  )  >''• 


P">  =-//,- 

P't.  =  '2  //.  4-  /  (  //J  4-  /'/,  )  /•". 

From   these  equations  and   the   fundamental    relation   M  (_//)- 
we  have 


-f-  /</ 

" 


PENTASPHERICA  L  ( :O()K DIN  ATES  2S5 

Ei't'rij  liiti'iir  t'tjittttion  <>f  tfif  tt>//»'  (-)  ;vy>/v.vr///'x  '/  xy<//»7v,  ///••  >'i/tt>r 
anil  tltr  rmltHK  <>f  ichii-Jt  <ir<-  </'('<'/<  /<>/  eyiuititinft  (4  ). 

It  is  convenient  to  represent  by  T;(")  the  numerator  of  /•'  in  (4j  ; 
thatis'  iW=.i?+«!+«*+<i*+,i?. 

We  have,  then,  the  following  classes  of  spheres: 
CASK   I.    ?/('/)  =r-0.    ynnsjM'i'idl  n]>Jn'n'8. 

Subcaw  1.  ?;  (  "  ) r-^-  0,  it ^  -\-  /V  -  i).  1'raper  sphrrat.  The  center  and 
the  I'adius  of  the  sphere  is  Unite,  but  neither  is  necessarily  real. 
The  sphere  does  not  contain  the  real  point  at  inlinitv. 

Siifii-nxt'  J.  TI(<I)  :-  0,  <i  +  i'i.~-  0.  (h'Jiniirii  i>lit>n'*.  The  radius 
is  infinite.  The  center  is  the  real  point  at  infinity.  Since  a  plane 
is  the  limit  of  a  sphere  with  center  receding  to  infinity  and  radius 
increasing  without  limit,  we  shall  call  this  locus  a  plane.  This 
may  be  justified  by  returning  to  the  coordinates  £,.  The  equa- 
tion then  reduces  to  ".,£., -f-  '*.,£,,-(-  ",£t~  "i^-, =  (>  w'ln  t'"'  condition 
<>';+  //.i'-f-  tif  ^-  0.  By  n'pctition  of  the  familial'  argument  of  analyti- 
cal geometry  this  may  be  shown  to  represent  a  plane. 

Since  this  case  differs  from  the  preyious  one  essentially  in  that 
the  coordinates  1:0:0:0:  /  now  satisfy  the  equation  of  the  sphere, 
we  may  say  :  .1.  y'r<y«r  jrfittn'  m<n/  /»'  defined  <t*  <i  nontpct'itil  sphere 
irJn'i'/t  jinn.*!'.*  tJir'iiii/h  flu'  r>'n/  point  "f  infinity. 

CASK   II.    ?/('/)=  0.    Special  »ji1n>rf*. 

Snb,;i^>  /.  7;('/)  =  "i  "  +''''."-'-  0.  /'nit if  xjihfrt'ft.  The  radius  is 
/.ero  and  the  center  is  not  at  infinity.  It  is  ohyious  that  the  sphere 
passes  through  its  center  //,-'',,  and  if  //,  is  real  the  sphere  eoii- 
tain>  no  other  real  point.  The  sphere  does  not  contain  the  real 
point  at  infinity. 

,S'////,v/.v,>  J.  7j(, /)—!),  //  -f-  >ii _  i._-  0.  SjH'i'itil  ji/iitii-ft.  The  radius  is 
indeterminate.  The  center  is  >r  :  ft.%:  n:  tt  :  itt,  which  is  a  point  at 
intinity.  The  e(|uation  of  the  sphere  may  be  written 


which,   in   ('artesian  geometry,   would  be  that    ot    a 

(^  s"  )•     In  this  case  the  sphere  contains  the  real  point    at 

Hence    yve    may    say:     .1    x/n'.-i>i?    /(/<>//<     /.<    -/    j>"int    ,<.•/<//•/••     ii'Jih-h 
p'lxxi-x  (lifi/ifi/h  thi-  /•.<//  jni//tf  "f  infinity. 


•JSti  TIIKKK    IHMKNSIONAL   ( i  K<  ).M  KTR  V 

The  locus  at  inlinitv  is,  as  we  have  seen,  ./•  -f  /./•.=  0.  Tliis  eonies 
uiiiK-r  Case  II.  Subcase  '2,  and  is  therefore  a  special  plane  with  its 
renter  at  1:0:0  :<);/;  that  is,  tin-  Im-iix  <if  ii/fuiiti/  IK  <t  *]>fi-i<i[ 
j<l,ni<-  >rJi»x>'  1-,'ttti-r  /.«  tin'  /•''<!/  i>"'nit  'it  iiifinlti/. 

119.  Angle  between  spheres.  The  angle  between  two  real  proper 
spheres  is  equal  or  supplementary  to  the  angle  between  their  radii 
at  anv  point  ot  intersection.  For  precision  we  will  take  as  the 
an^le  that  one  which  is  in  the  triangle  formed  by  the  radii  to  the 
point  of  intersection  and  the  line  of  centers  of  the  spheres.  If  0 
is  this  angle.  </  the  distance  between  the  centers,  and  r  and  /•'  the 

radii,  then  ,        .,        ,.,  , 

ir  —  r' +  r  '  -  _  rr  cos  v. 

If  now  the  equations  of  the  two  spheres  are 
y,/.r  =  0,  V/,.r  =  0, 

xW     '    '  s—l    '    ' 

an   easy  calculation  !>v  aid  of  formulas  (4),  ^11^,  and  (i!),  §117, 


whence 

C-OS0    =  "l'l   +   "-   ',-+'V.,+    "4    '4+"-     ':,  _  (1 


This  formula  has  been  derived  for  real  proper  spheres  intersect- 
ing in  real  points.  We  take  it  as  the  definition  ot  the  angle 
between  anv  two  spheres.  The  student  mav  show  that  if  one  or 
both  of  the  two  spheres  becomes  a  real  plane,  this  definition  of 
angle  agrees  with  the  usual  one. 

Tiro  .v////c/v.v  "V'/,./;   -~  (I.  "N  7<  ,-./•,.=  0  <rri'  nrili«<i<nml  H'/n'ii 

rtf,  _f_  „  /,  4-  „  /,  a.  „  /,  -f  a./..  =  0.  (2) 

It  both  of  the  spheres  are  iioMsprcial.  this  agi'ees  \\iih  the  usual 
detinition.  If,  howe\'er,  "S  n^-.—  0  is  a  special  sphei'e.  the  condi- 

tion    expresses    the     fact     that     the    center    of    "^  '',./',-      (l     lies    on     the 

shere  V/../-       o.     H 


'/'//'        //'    1-,'XHll  I'!/      'I/I'/     tllll'li'l,    lit      i-'i/lllit  I'l/l      tllilt      ll      Kill   I'Ull      ,vy<//(    /'(•      ftflnllhl 

/..     f,,'tJl'ii/"/Hll    f"    Hunt  In-  f   }<j>/l>ri'    /x   t/I'lf    t//>'   <;„/,,-    nf   tlf    *l>,'<-i'l1    vji/t>T>- 
11,       ,,,!     tin      ntJl.-r     •-'/'/"  I''  . 


PKNTASPIIKRK'AL  (  '(  ><  "HMUNATKS  li.sj 

EXERCISE 

Prove  tliat  the  coetlicients  '/,  in  the  e(|ii;itioii  of  the  sphere  are  pro- 
portional to  tlie  cosines  of  the  angles  made  l>v  the  sphere  with  the 
coordinate  spheres,  and  that  the  cosines  themselves  ma\  he  found  l>v 
dividing  ",  hv  V"i"  -f  "  ..*  +  "y"  +  -'4"  4-  ":?•  Compare  with  direct  ion  cosines 
in  <  'artesian  geometry. 

120.  The  power  of  a  point  with  respect  to  a  sphere.     If  r  is  the 

center  of  the  sphere 

2v,=o, 

with  the  radius  /•,  and  /'  is  any  point  with  coordinates  //,,  the  dis- 
tance ('/'  is  easily  calculated  by  (1),  $  1  1  s,  and  (»!),  ^117.  with 
the  result  : 


\\'e  shall  place 


=  r   •  _  r-  =  -       -  -•-         •  •          -  (.  , 

(",+    ''",-,)(//!+    '/A',   > 

and  shall  call  S  the  power  of  the  point  //,  \\'ith  respect  to  the  sphere. 
If  the  sphere  is  mil  and  the  point  //,  is  a  real  point  outside  the  sphere. 
the  power  is  the  square  of  the  length  of  any  tangent  from  the  point 
to  the  sphere.  If  the  sphere  is  a  point  sphere,  the  power  is  t  he  square 
of  the  distance  from  the  point  //,  to  the  center  of  the  sphere.  In  all 
other  cases  e<  pi  at  ion  (  '2  )  is  t  he  detinit  ion  of  the  power. 

From    ('2)   may   lie    obtained    the    important    formula    for   a    non- 
special  sphere  : 


The  above   discussion    fails    if    the   sphere    is   a    plane.      \Yc    may. 
however,  obiain  the  meaning  of  formula  (-\)  in  this  case  by  a 
process.     We  have,  from  ('_'), 

where  /'.I  is  the  shortest   distance  from   /'  to  the  sphere.     Then 

,s'  /  /'< ' 

r  '    /' 


288  THREE  DIMENSIONAL  GEOMETRY 

Now  let  (''  recede  to  inlinity  along  the  line  /'('.  The  sphere 
becomes  a  plane  perpendicular  to  /'./.  Hut  the  limit  of  -  —  •  as 

r   becomes   infinite   and    ''j  +'*''.,   approaches   zero,    is   1,   from    (1). 

Therefore  s- 

Limit-  =  -2  I1  A, 
r 

where  /'.I  is  the  perpendicular  from  /'  to  the  plane.  This  result 
mav  be  checked  bv  replacing  j\  by  £,  and  using  familiar  theorems 
of  Cartesian  geometry. 

The  equation  of  any  nonspecial  sphere  mav  be  written  so  that 
i](<t)=\.  The  equation  is  then  said  to  be  in  its  normal  f<  inn  ^  and 
the  denominator  <i{+  '/.;-f  <ij+  <i;  +  a:  disappears  from  equation  (-}  ). 

121.  General  orthogonal  coordinates.  Let  us  make  the  linear 
substitution 

p.r[  =  atl  r,  +  a,  .,>•._,  +  ni3j-^  +  n  ,4.r4  +  a  ,s.r5,      (  /  =  1  ,  -,  :J,  4,  .">  )       (1  ) 

in  which  the  determinant  .ncik\  does  not  vanish.  Then  to  any  set  of 
ratios  j\  corresponds  one  set  of  ratios  r(,  and  since  the  quantities  .r. 
satisfy  a  quadratic  relation  tu  (./•)  =  (),  the  quantities  ./-,'  satisfy  another 
tjiiadratic  n-lation  O  (./•')  =  0. 

Then  values  of  ./;  which  satisfy  H  (./•'')=  0  correspond  to  one  and 
only  one  set  of  ratios  of  ./•.  wliicli  satisfy  <u  ('./•)=().  Therefore  ./•' 
can  be  taken  as  coordinates  of  a  point  in  space  and  are  the  most 
general  pentuspherical  coordinates. 

The  sphere  V^r,  =  0 

becomes  the  sphere  V",'-'','  =  "i 


and  the  condition  ?;('')=  0  for  a  special  sphere  goes  into  another 
quadrat  ic  condit  ion  1  1  (</'  )  =  (|. 

The  point  at  inlinity  takes  the  new  coordinates  '(,,  4-  i'l,-,,  and  the 
condition  that  a  sphere  should  be  a  plane  is  that  its  equation  should 
be  satisfied  by  these  coordinates. 

The  coordinates  £,  of  ^117  furm  a  special  case  of  these  general 
coordinates.  \\V  >hall  not.  however,  pursue  the  treatment  of  the 
'_!viirral  case,  but  >hall  restriet  ourselves  to  the  case  in  which  the 
ti\e  eoi'trdinate  spheres  are  orthogonal.  In  thi<  case  no  sphere  can 
be  -pn-ial.  since,  it  it  were,  its  center  would  lie  on  each  of  the  other 


PKNTASIMIKHH'AL  molUHNATKS  :>'.) 

four  split-res,  and  then-  would   lit-  four  orthogonal  spheres  through 
a  coiiiinoii  point,  which  is  obviously  al>surtl. 

\\'t-   niav  consider  that   cadi  of   the   equations   nf  the   coordinate 
spheres  has  heeii  put    in   the  normal   form,  >o  that  \\'e  have,  in  (  1  ), 


Then,  bv  (  o  ),  ^  1  JO,  the  substitution  is  expressed  bv  the  equations 

pj'[--    '''<  (4) 

/•, 

where   N    is   the   power  of  the   point    j\    with    respect    to   the   sphere 

.) 
./•'  =  0,  and  /•    is  t  he  radius  of  ./•'  =  <>   since  the  factor  is  the 


(  'onset  pit-lit  Iv  \ve  have  for  ./''  the  tninlamcntal  relation 

./'[-+  XH  J\.~  +  .r(-    f   ./•'"-,    0,  (S   , 

and  the  condition  lor  a  special  sphere  is 


15  y  (  1  ),  $  11s,  the   ratlins  /•'  nf  the  sphere.'-'       (l  i 


•J'.)0  THKKK   IHMKXSIOXAL  (1KUMKTKV 

where,  if  anv  sphere  .i\  -"  is  a  plane,  the  corresponding  coordinate 
./•;'.  is  /ero,  as  in  fact   happens  when  rk=  ~s.. 

The    equation    ./^  4-  /./'.  ==  0    tor    the    locus    at    infinity    becomes, 
from   (  in  )  and   (11). 


where,  a'_rain,  if  anv  coordinate  sphere  is  a  plane  the  corresponding 

term  vanishes  I  rom  (  1  ;>>  ). 

It    is  now  easy  to  see  that   the   formula  (  ti  ).   vj  1  1  7,   for  distance 


4.  .,  ^  4.  _r 


V 


so  that  the  equation   of  a  sphere  with  center  //,  and  radius  /•  \- 


Identifying  this  with         "^  '/'./,'      " 

We  have  p<i\  =  //',+ 

]-"roin  (11).  with  ( •> )  and  ( ."> ). 

v ' , 


S'~''i       XT//! 
;=-,- 


(115) 

(17) 

(18) 

(  1 !» ) 


I'>\-    si|iiarinur    (17).    adding,    and    i-cducin^    hy    (  *  ).    (  1  <s ).    and 
<  T.t  ),    we   obtain    the    following    formulas    for   the    radius    and   the 

center   of   the   sphere   (  1  ii ) :         _  , 

/  ii"2 


V 


r  ,-*<!, 

vi/,     -  </,  7 

-  /',  ""™  /'. 

The  formulas  ot    fj  1  1  s  are  oiilv  special  ea>es  ot   these 


EXERCISES 

1  .     1'p  iVe   the    I'elal  '.'  'Ii    "V  L'. 


v,/,,. 


I'KNTASlMIKKirAL  COORDINATES  2!»1 

122.  The  linear  transformation.    ('onsidera  linear  transformation 

£>•''!  ""',  I    ''l  +  'f,J-''j+   ",.;•'':  +  ",l-'4  +    'li:,  •'':,'  <    1    ) 

in  which  the  determinant  -i,,.  does  not  vanish  and  l>v  which  the 
fniidanicntal  relation  o>(./-)-=0  is  invariant.  Then  tin-  relation 
;y(  ,f  )  _-.  (I  is  also  invariant. 

'1  he  relations  (1)  define  a  one-to-one  transformation  of  space 
1>\'  which  a  nolispeeial  sphere  i^'oes  into  a  non>pecial  splicrc  and  a 
special  sphere  into  a  special  sphere.  There  are  two  tvpes  to  he 
di>t  in^tiished. 

I.  Trim*)  "Tiii'it  tons  f>ij  irlifli   tin'  /•>'i/  !>'>////   lit   i/itiiut/i  /,•<  uii'ii  rut  nt  . 
I»v  such  a  t  ranst'oi'inat  ion   planes  are  transformed   into  planes  and. 
consequently,   straight    lines    into   straight    lines.     Since    the    trans- 
formation  is  analvtie   it    is  a  coilineat  ion. 

Point  spheres  ai'e  transformed  into  point  spheres;  therefore, 
expressed  in  ('artesian  coordinates,  the  transformation  is  one  hv 
which  minimum  cones  <_n>  intu  minimum  cones,  and  consequently 
the  circle  at  intinitv  is  invariant.  Hence  the  transformation  is  a 
met  rical  t  raiisformat  ion. 

(  'onversclv,  anv  metrical  transformation  mav  he  e\pre»ed  as  a 
linear  transformation  of  pentaspherical  coiirdinates.  This  is  easih" 
seen  hv  use  of  the  .special  coordinates  of  >  117  and  is  consequently 
true  for  the  general  coordinates, 

Hence  '/  lutriir  t  r<i  nxt'nrnnit  i«n  <>t'  jn'nfil*j>/n'i'{f<tl  cin'iri  li/mti  x  /-_// 
/r/iii'/i  tin'  /••  -ill  jm'tiit  nt  iiifunti/  /x  i  ni-iiri'i  nt  />•  a  ///,  trc'iil  t  r<i  iixt'"rimit  i"n, 
itii'l  f«n  i'i  i'*it  i/. 

II.  Ti'dtiftfui'/iHitl'inx  I*'/  i/'hii-li  llir  r'lil  ji'iint  til  (iilinilii  ix  i/"/  ///>'•!- 
I'inif.     Ainoii^'   the>c   transformations   are   the    fni'i-/'m"ii#.      I  hat    an 
inversion    mav  he   I'eprcsentetl   actually   1)\    a    linear  t  ran.-tormat  ion 
of  peiitaspherii-al   coi'inlinates    i--   evident    from   the   example   in    the 
coi'/rdmates  ^,  vj  1  1  7.  t<       /.-<t 


p?, 


1>'.»1>  THKKK   DIMKNSIONAI,  (IKOMKTKV 

Consider  now  the  general  case  of  a  real  transformation  by  which 
the  real  point  at  intiiiitv  /  is  transformed  into  a  real  point  J.  and 
tlu1  same  point  .1.  or  another  point  J',  is  transformed  into  /. 
Sinee  the  transformation  is  real  .1  cannot  he  at  inlinitv.  Let  this 
transformation  be  7' ami  let  .S' be  an  inversion  with  ./  as  the  center 
of  inversion.  Then  the  product  .s"/' leaves  /  invariant  and  is  there- 
fore a  metrical  transformation,  .'/.  Therefore  A"/'  =  M ;  whence 
7'=N  'J/.  I  Jut  N  '  =  .S'.  Therefore  7'=.S'.J7.  llciice 

Anif  /•>/(/  tritnxfurnmtiun  of  pfttt<ti<j>}icri<-<il  <•>, </'/•<////< it, 'a  li/  //7//<7<  tin' 
/•lit/  [i"int  tit  infinity  ix  >i"t  incii/'Kiitt  ix  citht'i'  tin  iiit'i'mivn,  <>r  (//<' 
l>r.,,lii<'f  i if  tin  titi'f/'xiit/t  mi'J  <i  met ri'-iil  tr<tnxfi>i'in<tti>n. 

This  does  not  exhaust  all  cases  of  imaginary  transformations. 
\\'e  may  obviously  have  imaginary  transformations  of  the  metrical 
tvjie  or  inversions  from  imaginary  points,  so  that  the  above  theorems 
hold  for  transformations  by  which  the  real  point  at  inlinitv  is  trans- 
formed into  itself  or  into  any  finite  point.  Transformations,  however, 
by  which  the  real  point  at  infinity  is  transformed  into  an  imaginary 
point  at  infinity  are  of  a  different  type.  An  example  of  such  a 

transformation  is        , 

'       -    '  -  ~     ' 


PJ:     .,•'•> 

p.r'.----    '  x^    -  -i.r..      -  -.,' 


PJ   =       -'\ 

We  shall  close  this  section  \vith  the  theorem,  important  in  subse- 
quent work:  If  t1t>-  ruilrdiit'itt'  xi/st, •///  in  nrt/i'ii/innil  tJi<-  1r<tnxf<,rin<t- 
fi"/t  f.rfi/; -x.v»  •/  /-//  rJuinifimj  ///»•  >•/////  '_-/'  i>nc  «f  f/i>-  rnHril.iitnti-x  ix  (tit 

in  I'f/'x/nit     an     tfi,'    I'n/'i'i  -!<jiii/t'l/>i</    i'<i<ii-ii  t  n<it,'    ,vy;/«'/V. 

l-'or   let    the   si'j;n    of  j'k  be   changed.    Then    points  on   the  sphere 
n  are  unchanged,  and  any  sjihere  ortlio^onal  to  ./,      n  is  trans- 
formed into  it>elf.    This  characteri/.es  an  inversion  un  .rk  =  n. 

EXERCISES 

1.    Trove  the   laM   theorem  aiiah't  ic;illv.  u--iiit,r  the  turmulas  of  •;  ll'l. 
12.    Trove    thai    the    jiroduet    of    Jive    inversions    with    respect    to    live 


PENTASPHEKICAL  COORDINATES  'J«i;j 

123.  Relation  between  pentaspherical  and  Cartesian  coordinates. 
If  we  take  the  axes  <>.\\  »}  .  <  >'/.  Used  in  xj  117  In  define  the  speeial- 
i/.ed  pent  asjilit-i'ical  coordinates  as  the  axes  also  ol  a  set  ol  (  'artesian 
coordinates,  it  is  obvious  that  \\  e  have,  for  real  points, 

p.'\  =  •'•"    -f  //"  +  ?~  ~  1         =  •'•'  +  if  -f  Z"  -   <  ~, 

/j.r,=  L'./-  L'./Y, 

/>.'',  =    -  //  -  //'•  (  1    ) 

rt    ,.__•)      -5  _       •)      -.  t 

- 


This  establishi'S  in  the  lirst  plaee  a  one-to-one  eorre.vpondenee 
between  real  [mints  in  the  two  systems.  It  may  be  used  also  to 
define  the  eon'opoiideiiee  between  the  inia^iiiarv  and  infinite  points 
introduced  into  each  system.  There  exists,  however,  no  reason 
why  such  points  introduced  into  one  system  should  always  ha\e 
corresponding  points  in  the  other.  As  a  matter  of  fact  a  failure  of 
correspondence  of  such  points  does  exist. 

The  (  'artesian  points  on  t  he  imaginary  circle  at  infinity  fail  to  exist 
in  pentaspherical  coordinates  since  values  of  ./•,  //.  .;•,  /  \\  hich  sat  i>t  \  the 
relations  /"4-//"+  r^  ",  /  =  0  ^ive  j'}  :  ./-..:./•.:  .>'t  :  r,=  0  :  0  :  0  ;  0  :  n. 
iiut  anv  ('artesian  point  at  infinity  not  on  the  imaginary  circle 
corresponds  in  pentaspherieal  eoi'irdinates  to  the  real  point  at 
i  11  1  i  it  i  i  v  1  :  0  :  0  :  0  :  /. 

(  )n  the  other  hand,  we  ha\e  in  pentaspherical  ^voinetrv  inui^inarv 
points  at  in  tin  it  v  satisfying  t  he  re  la  t  ions  j\*+  .r:  +  ./'~  —  0,  ./^  -f-  /./•._  =  i), 


in  ( 'artesian  uvoinet  rv  since  no  values  of  ./•  :  //  :  z  :  t  in  (  1  )  ^ix'e  them. 

This  failure  in  the  correspondence  is  of  importance  if  one  wi>hes 
to  pass  from  one  system  to  the  other.  'I  hey  are  ol  110  significance-, 
howevel',  as  lon^  as  one  operates  exclusivulv  in  one  >\>tem. 

The  general  pentaspherical  coi'irtlinates  are  connected  \\iih  ('ar- 
tesian eoi'irdinates  b\-  equations  ol  the  form 

p.r[  =  ( 't,,  -f  '''(,-,  )  (  J-"  -f  if  f  2" )  -f  -  'i. ._..''  +  -  ",  //  f  -  ",,~    -  <  "  i      ''",    )• 

124.  Pencils,  bundles,  and  complexes  of  spheres.  If  ^/',':  "  ;"ul 
^  l>j\  =  0  are  two  spheres,  ihe  eijiiation 

V(  „    f  \/,  )r  .     (I  (  1  ) 


1>!»4  TH1IKK-  DIMENSIONAL  GEOMETRY 

represents  a  sphere  through  all  points  common  to  the  two  spheres 
and  intersecting  neither  in  any  other  point.  Such  spheres  together 
form  a  j»  m-il  of  spheres. 

.1  fi  ii'-U  »''  ttji/nTex  <-<>nt<tinx  one  <tn<J  onl//  one  ]>l<tn<'  unlexx  it  <'"ii- 
.v/.v/.v  i  ntirflii  "f  l>I<iiti  x. 

This  follows  from  the  t'm-t  that  the  condition  that  e([iiatioii  (  1  ) 
should  he  satisfied  l>y  the  coordinates  of  the  real  point  at  infinity 
consists  of  an  equation  of  the  first  decree  in  X,  unless  both 

?  <i  j:  =  0  and  "N  /»./•  =  0  are  satisfied  by  those  coordinates.    In  the 

—    '  ^—t 

latter  ease  all  the  spheres  (1  )  are  planes. 

A  /»/!/•//  lit'  Hjthi'n'if  ci>nt<tinx  ttc</  dtnl  onli/  ttco  ajiecidl  uphere*  (  which 
nini/  ?>f  ri'<</,  i/inii/i  inii'if,  './/•  coincident)  unlt'tsy  /(  I'vnxtxtx  entirely  <>f 
XJH  i'iii/  npnefi'K. 

The  condition  that  (1)  represents  a  special  sphere  is 
»;(</  +  X/-  )  =  T/(//)  +  X?;  (  <i,  I  )  +  \~jj  (/<)-=<>, 

which  determines  t  wo  dist  inet  or  equal  values  of  X  unless  ?/(,/)=  (), 
'/(M  --  -  (|.  ?/  (  ".  f')=0.  The  latter  case  occurs  when  the  two  spheres 
N  (/_./•_  =  (I,  N  /,./-|=z  0  are  special  spheres  with  the  center  of  each  on 
the  other. 

The  theorems  of  J^  111  and  others  analogous  to  those  of  ^  tli*  are 
easily  proved  by  the  student. 

If  V,  /_./•  —  0,  ^£,I>fi\=  0,  '5yV*'i-==  0  are  three  spheres  not  in  the 
same  pencil,  the  equation 


represents  a  bundle  of  spheres  as  in  ^  1  1  L*.  The  bundle  contains 
a  Mildly  infinite  set  of  plant's  and  a  singly  infinite  set  of  special 
sjiheres.  The  relations  between  orthogonal  pencils  and  bundles 
found  in  ^  1  1  '1  are  easily  verified  here. 

If  V,/.,-  _:  0.    V/v7',-^  °-    V'V,—  0,    V  «/,.'',  =  0    are   four   spheres 
not    belonging   to   the   same    bundle,  the   equation 

V(",  -f  X//.4-  fn't+  V(lt)J\=  <> 

represents  a  complex  of  spheres.  It  consists  of  spheres  orthogonal 
In  a  base  sphere  and  contains  a  doubly  infinite  set  of  planes  and  a 
doubly  infinite  set  of  special  sphere.-.  'I  he  centers  ol  the  latter 
form  the  base  sphere. 


PENTASPH ERICA L  COORDINATES  U'j:, 

EXERCISES 

1.  Prove  that   the  angle  under  which  a  sphere  cuts  any  sphere  of  a 
pencil  is  determined  l>y  the  angle  under  which  it  cuts  two  spheivs  of 
the  pencil. 

2.  Prove   that    among   the   spheres   of  a    pencil    there  is  ahvavs   one 
which   cuts  a   given   sphere   orthogonally. 

3.  Prove  that   the  angle  under  which  a  sphere  cuts  anv  sphere  of  a 
bundle  is  determined   l>y  the  angles  under  which   it  cuts  three  spheres 
of  the  bundle. 

4.  Determine  a  sphere  orthogonal  to  lour  given  spheres. 

5.  Determine  a  sphere  cutt  ing  live  given  spheres  under  given  angles. 
\\heii  is  the  problem  indeterminate'.' 

125.  Tangent  circles  and  spheres.  Let  //,.  .?,,  /,  be  anv  three 
points  o'iven  in  orthogonal  pentasplicric-al  coordinates,  and  consider 
the  ^nations  ps.=  1/^X2.+ pf;.  (1) 

In  order  that  J\  should  be  the  eoi'irdinates  of  a  point  it  is  neees- 
sarv  and  sut'tieieiit  that 

^ (//,.+  X.rt +  K )--  0.  (l>; 

Since  N  _,/--.  ii,  "V  :••-=<>,  ^/^.— I),  equation  ('!)  reduces  to 

.IX  +  /•>  +  r^/^  =  (J,  (o) 

\\here  A  -     ^ .//,.=',,   11=  V.'//,,   ('       V;'/,- 

Therefore  (  1  )  inav  be  written 

.IX 

P->\    ://,-+5Ur,—  ,,',-,  ',-'  <4' 

/•  -p-  i  x 

or  p.t\  —  /.'//,  4-  ( ' '//,  +  /'X  -  .  I/1, )  X  +  <  '-r,X'J.  ( •"> ) 

'I  his  represents  a  one-dinieiiMonal  extent  ot  points.  An\  sphere 
which  contains  the  three  points  //^  *..  ft  \\ill  also  contain  all  the 
points  ./• .  and  anv  point  ./•,  belongs  to  all  the  spheres  t  liroii^h  /^,  »<(  fr 
Therefoi'c  (  t)  re[)re>eiits  a  ciri-li',  including  the  special  case  of  a 
straight  line. 

Any  e<|iiatioii  /(  »•  ,  ./-.,,  ./'..  ./'(,  ./•.  )  - --  0,  (l!) 

where^  is  a  lioiun^i'licuits  jiolviminial  ot  the  /t{\\  decree,  represents 
a  surface.  To  !'md  where  it  i->  cut  hv  an\  circle  substitute  from 
( •>  )  nit  o  (  (1  ).  There  res  11  It  s  an  e<|iiat  ion  ot  derive  l!  /<  ill  X.  so  t  hat 
the  surface  i>  cut  1>\  an\  i -uvle  in  'In  iioints. 


l>'.Hi  THREE   DIMENSIONAL  GEOMETRY 

If  ('artesian  coordinates  are  substituted  for  .;-,  in  ((>)  the  equation 

is  of  the  'J//th  order  and  of  the  form 


where  HI  is  u  homogeneous  polyiioiniul  ot  decree  k  not  containing 
(  ./-"-f-  _//-+,  r)  as  a  factor.  The  surface  therefore  contains  the  circle 
at  inlinitv  and  as  an  /Mold  curve  it  nt  -*-  0.  In  the  ('artesian 
geometry  the  surface  is  cut  l>y  any  c-irele  in  4  //  points,  hut  the  cir- 
cular points  at  infinity  count  -  n  times  and  do  not  appear  in  the 
tet  racvdical  ^'eonietrv. 
The  equation  in  X  is 


,,  //.,.  ,//.,,  //(  )  +  \H"    !  V  '•    (<  ;//_  -f-  /;2.  -  .  j/( 


u.      7 


Xo\v   if   //    is  on  the  surface,  then    /'(  //)  =  0  and   V  //(    •      =  (),  the 

~*  "    '  //, 
latter  hecause  /'is  hoino^tMieous.     Thei'ctore  one  root  ot  (7)  is  /ero. 

Two   roots   will   he  /.en>  it,    in   addition   to  //i  being1  on  the  surlai-e, 
\\'e   ha\  e 


which  is  the  same  a> 


It  this  condition  is  satistied  hv  the  t\\o  jioints  ?_  and  /,,  the  circle 
(  1  )  is  tangent  to  the  surface  (  ti  >  at  //,.  The  condition  is  certainly 
met  it  r(  and  tt  ai'e  hoth  oil  the  same  sphel'e  of  the  [icllcil 


Anv  sphere  of  this  pencil   has  accordingly  the  propertv  that  an\' 
plane  section  of  it  through  //,  is  a  circle  tangent  to  the  surface  ( t! ). 

Therefore   (  '.'  )    fi/'fi'Xi'tltK  «  />fi«-i/  nf  t,(Hi/rnt   X/Jirrrx   I"   tin    xu/'fitn; 

It     '         u.  all  circles  throii'di    //    meet  the  surlace  in   two  coinci- 


PENTASPHERH.'AL  COORDINATES 
126.  Cyclides  in  pentaspherical  coordinates.    (  'onsider  the  surface 


From  §  l'J:>  ;uul  £  llti  this  is  a  cyclide.  We  have  shown  that  if  the 
cvdide  has  singular  {mints,  it  is  tin-  inverse  of  a  quadra-  surface. 
We  shall  therefore  limit  ourselves  here  to  the  general  i-ase  in  which 

the  singular  points  do  not  exist.     Since,  then,  the  equations    •     =  0 

'.'/, 
have   no   common   solution,    it    is   necessary   and   suHieient   that   the 

discriminant    <tik    does  not   vanish. 

It    is  a  theorem  of  algebra  that    in  this  case  the  quadratic  form 
mav  he  reduced  bv  a  linear  substitution  to  the  form 


(where    <\  '^-  0  ),    at    the    same    time   that    the    fundamental    relation 

a)  (  ./•  )  is 

j--  +.>::+  x-  +  .>•-  +  j--  =  0.  ('.}) 

We  shall  therefore  assume  that   the  etjuation   of  the  eyclide  is  in 
the  form  (  '2  )  and  that  the  coordinates  are  orthogonal. 

From  equation  (~2)  it  is  obvious  that  the  equation  of  th*-  surface 
is  not  altered  bv  eluuiging  the  sign  of  any  one  of  the  coordinates  j-. 
Hut  this  operation  is  equivalent  to  inversion  on  the  sphere  ./•,--<!. 
1  It-nee 


The  {>eiicil  of  tangent   spheres  to  the  cvdide  at  any   point    tt  t  is, 

bv  ^  i-j:.. 

^('V+\)//ij-.=  0.  (4) 

Hence,  ill  order  that  a  given  sphere 


should  be  tangent  to  (  -  ),  it   is  necessary  and  sut'licieiit  to  determine 
/\  and   //    so  that 


ii       V  0.      V    "'         o.  (7) 

—  (  ••   f-  X  )'J  — ••_  I    \ 


208  THREE-DIMENSIONAL  GEOMETRY 

of  which  the  first  is  a  consequence  of  the  last  two.  The  last  two 
express  the  fact  that  the  equation 

y~^-  =  0  (8) 

Av+X 

lias  equal  roots.  This  imposes  a  condition  to  be  satisfied  in  order 
that  (  ">  )  should  be  tangent  to  (-). 

When  X  has  been  determined  from  these  equations,  equations  (0) 
determine  //,  in  general  without  ambiguity.  Exceptions  occur  if 
X  —  —  ck,  where  i-k  is  any  one  of  the  coefficients  of  (-).  In  that  case 
we  have  in  (li)  ak  =  0,  and  yk  cannot  be  determined  from  (»>).  How- 
ever, if  the  other  four  coordinates  //,  are  determined,  t/k  has  two 
values  of  opposite  sign  but  equal  absolute  value,  determined  from 
the  fundamental  relation  ('••>),  The  corresponding  sphere  (•">)  is 
orthogonal  to./-;.=  0  and  tangent  to  the  cyclide  at  two  points  which 
are  inverse  with  respect  to  j-k  —  0. 

The  value  of  X  may  be  taken  arbitrarily  as  —  <-k;  whence  <ik  =  0. 
The  values  of  <tt(i^k)  must  then  be  determined  from  (7)  with 
\=—ck.  Each  of  the  tirst  two  equations  contain  an  indetermi- 
nate term.  The  last  equation  becomes 

£    "'      =  <>•          (,>*)  (9) 

i  '  .     '  t 

The  coetlicients  of  (  f>  )  satisfy  two  equations,  therefore,  and  the 
spheres  form  a  family  of  spheres  which  is  not  linear.  In  this  family 
a  sphere  can  be  found  which  is  tangent  to  the  cyclide  at  any 
given  point.  For  it  X=  —  <\.,  and  //,  is  any  point  on  the  cvclide, 
equation  (ii)  will  determine  <tt,  and  the  <//s  will  satisfy  ('.<).  as 
has  been  shown.  The  spheres  of  the  family  therefore  envelop 
the  cyclide. 

There  are  five  such  families  of  spheres,  since  X  may  be  any  one 
of  the  live  coefficients  <•..  Hence 


Tin  i/i'/irfil  '•//'•//'/-•  /*•  <'n>'<-l«f><-<l  lii  fii-,'  j'uniilif'x  lit'  ,vy*/^'/vx,  f 
lniily  i-i.inxixttinj  lit'  8pln'rt>»  vrthiH/vtuil  tn  «/H'  «J  thi-  (iff  c'nofiUn 
//t7v,v  anil  f<ini/i'/if  t<>  tin-  stirfai-c  lit  tu'u  fioi/itx. 

\\'e  >hall  sliow  that  t)t>-  i-cntcr*  <>f  thf  i<pJtt-rt'n  <.>t'  fti-Jt  .sv/vVx  //»• 


PENTASPHEUICAL  <  <><  >KI>IN  ATKS 

Take,  for  example,  the   series  for  which  X  =  —  <•    and  <i  —  o.     If 
//c  are  the  coordinates  of  the  center  of  a  sphere  of  the  series  hv  ('J(»), 


_fv 


and  (iic= 

whence 

and  equation  (0)  becomes 


y  _<  .V>  -  .y, 

^r1    r,  (  •-,  —  <• 


whieli  is  the  equation   of  the  locus  of  the  centers  of  the  spheres  of 
the  family  under  consideration. 

By  (4),  §  llM,  equation  (10)  may  be  written 


yOV-'SV^o. 


(11) 


and,  finally,   if   X;.  and    .s'j  are  expressed  in   Cartesian   coordinates, 
equation  (11)  is  of  the  second  degree,  and  the  theorem  is  proved. 
We  mav  sum  up  in  the  following  theorem: 

Tit''  </i'ntT>/7  i-i/i'llijf  until  Lc  f/f/n  ritti'J  In  fir,'  inti/x  iix  f/t,'  ,'n>->'!«jh>  nf 
a  x/i/n  /•>•  aiilijfi't  t<>  f//t'  f'i'o  fii/tififiiniK  t//'/f  it  xhnnltl  I"-  i>rth"</ii>i'tl  to  <t 
fi.n'il  Hither?  mill  t}mt  ft*  ,-,-nt,T  xjmitld  //»•  on  ,i  ij>nttln'<'  xnrt'iiC''. 

A  surface  which  is  its  own  inverse  with  respect  to  a  sphere  .S' 
is  called  tinnllutimntw  with  respect  to  \  \\hirh  is  called  the  .///•-<•- 
f  n'.r  .v/'//ov.  Such  a  sui'face  is  enveloped  l>v  a  familv  of  spheres 
orthogonal  to  N  and  doiihlv  tangent  to  the  surface.  Fur  at  anv 
point  /'  of  the  surface  thei'e  is  a  sphere  tangent  to  the  surface  and 
orthogonal  to  \  !>v  inversion  this  sphere  is  unchanged.  It  is 
therefore  tangent  to  the  surface  at  /''.  the  inver>e  of  /'. 

The  surface  on  which  the  centers  of  these  enveloping  spheres 
of  the  auallamatic  sui'face  lie  is  called  the  i/i-t'i-r<  nt. 


300  THRKK-  DIMENSIONAL  (JKOMKTKV 

EXERCISES 

1.  If  <}t.  is  (iiu-  of  tlic  five  (left-rents  of  the  eyelide.  and  \  the  corre- 
sponding directrix  sphere,  prove  that,  the   tetrahedron    whose   vertices 
are  the  centers  of  the  other  tive  directrices  is  self-conjugate,  both  with 
respect  to  <4  and  with  respect  to  St. 

2.  Prove   that   on   the  eyelide  there  are  ten   families  of  circles,  two 
families  corresponding  to  each   ot    the   live   modes   of   generating   the 
eyelide. 

3.  The  focal  curve  of  any  surface  being  defined  as  the  locus  of  the 
centers  of  point  spheres  which  are  doubly  tangent  to  the  surface,  prove 
that  the  eyelide  has  live  focal  curves,  each  being  a  sphero-quadric  formed 
bv  the  intersection  of  a  deferent  by  the  corresponding  directrix  sphere. 

REFERENCES 

For  intire  reading  along  the  lines  of  I'itrt  III  of  this  bonk  the  following 
references  are  given.  As  in  Part  II,  these  are  imt  intended  to  form  a  complete 
bibliography  or  to  contain  journal  references. 

ficntrnl  trt  utixi  x 

( '[.r.ii-oii-LiM>i.M AN.  Vorlesungcn  liber  (ieonietrie.    Tenbm-r. 

DAUimrx.    See  reference  at  end  of  Part   II. 

Ni  I.WKM;  I.OWSK  I,  <  ieonif't  rie  dans  l'es|iaee.    ( iant  hier- Villar>. 

S  v  i.MoN-Koi.Kus,  (ieometry  of  'I'hn-e  Dimensions.     Longmans,  (ireen  \  Co. 

( ';>'•/<   a tvl  spturi's  : 

Cooi. UK, i:.  Circle  and  Sphere-  (ieometry.    Oxford  Clarendon  Press. 

( >•//</»  x  : 

BOCHKI:.  Potentialtheorie.    Tenbner. 

DAKHOI  \.  Snr  nne  elasse  ri'inanniablo  de  conrbes  et   de  sin-faces  ali,rehri<iues. 

A.  Hermann. 
Dni-.m.KM  \NN.  (ieoinet  rise  lie  Transfonnat  inn  en.  1 1.  Teil.    ( iiischen'M-he  VerhiLrs- 

liandlnnir. 

'I'hese  are  in  addition  to  the  ireneral  treati.-es  of  Darlioiix  and  Coolid^e  already 
refei-red  i,,.  The  tir.-t  section  of  l'>ocher's  honk  is  of  intere.-t  here.  Doelilemann 
contains  figures  of  the  Diipin  eyclides. 

.  I  li/i  hriiif  njii  rut  inn*  : 

P>oi  III.K.  Introdiii'tion  to  Higher  Algebra.    The  Maemillan  Comiiany. 
MKOMWK  n,   (Quadratic    Forms  and   their  ( 'lassiticat  ion   by    Means  of    Inviriant. 
Factors.    ( 'ambridire  I'niversitv  1'ress. 

l-'.acli  of  these  books  contains  ueomet rical  illustrations.  The  >»ndent  mav  refer 
to  till-in  for  any  algebra!''  ni'Mhods  \ve  ha\e  employed  and  e>pr,-i;il] y  for  an  expla- 
nation of  the  method  ot  elementary  divisors  in  rediicinL'  one  or  a  pair  of  quadratic 
form*  to  \arions  type>.  Uronr-vich  contains  a  full  cla.-silication  of  the  eyclides. 


PART  IV.    GEOMETRY  OF  FOUR  AND  HIGHER 
DIMENSIONS 

(TIAPTKR   XVII 

LINE  COORDINATES  IN  THREE-DIMENSIONAL   SPACE 

127.  The  Pliicker  coordinates.  The  straight  lines  in  spare  form  a 
simple  example  of  a  four-dimensional  extent,  since  a  line  is  deter- 
mined l»v  four  coordinates.  In  fact,  the  equations  of  a  line  can 
in  general  be  put  in  the  form 

*•=«  +  ,, 

//  =  KZ  +  a, 

and  the  quantities  (V,  *,  /?,  a )  mav  he  taken  as  the  coordinates  of 
the  line.  .More  symmetry  is  obtained,  however,  by  the  following 
device. 

From  ('([nations  (1  )  we  have 

>•//  —  .v.r  =  rrr  —  p*,  ('2  ) 

and  we  mav  place  rv  —  px  =  ?;.  (  •> ) 

thus  obtaining  live  coordinates  connected  bv  a  quadratic  relation. 

If  (./•'.  //',  z  )  and  (./•",  //",  z" )  are  anv  points  on  the  line  (1  ).  we 
mav  casilv  compute 

/•:  x:  p\  rr:  i/ :  }  --.  jc    -./•":_//'  -  >/" :.r"z' —  .r'z" :  i/"z'  -i/'z":j''i/"  —  j''t/':z—z", 

and  it  is  the  ratios  on  the  right-hand  side  of  this  equation  which 
were  taken  bv  1'liieker  as  the  coordinates  of  a  line. 

These  coordinates,  however,  form  onlv  a  special  case,  arising 
from  the  use  of  ('artesian  coordinates,  of  more  general  coordinates 
obtained  bv  the  u.se  of  quadriplaiuu'  coordinates.  \Ve  proceed  to 
obtain  these  coordinates  independently  of  the  work  (list  done. 

'I  he  posit  ion  of  a  st  raight  line  is  tixed  bv  t  wo  points  (  ./•  :  ./-.,:./•.,: r  ) 
and  (  //t ://.,:  //.,:  //,  ).  It  should  lie  possible,  therefore,  to  take  as  coor- 
dinates of  the  line  sonic  (unctions  ol  the  coordinates  ot  these  t  \\  o 
jiomts.  Furtiierinore,  .since  anv  two  points  \\lmse  coordinates  are 


F<>ri;    DI.MKNSIONAL   (JKO.MKT1IV 

X.r  +  fiv  mav  be  used  to  define  the  same  line  as  is  defined  bv  .r 
and  //,.  the  coordinates  of  the  line  must  be  invariant  with  respect 
to  the  subst  it  ut  ions 


p.r[  =  X^r  -)-  /MJ//,,  p;i\    -  \.  ,r  -f  //.,//,.. 

Simple    expressions    fulfilling   these   conditions   are    the   ratios  of 

determinants  of  the  form     '  '  '  '    .     \Ve  will,  accordingly,  consider 

i  •  \J\-    "i 

the    expressions 

Pit—  -r^t-  Wi- 

Since  pit=    —  /»,,.,  there  are  six  of  these  quantities;   namelv, 


n    —  .r  n    -  ./' 

/    14  1-4  4- 


;>.,,=  .'y/.  —  . 

which  are  connected   bv  the  relation 

,-  I 


//,      //,      //,      ,'/4 

It  is  obvious  that  to  any  straight  line  corresponds  one  and  onlv 
one  set  of  ratios  of  the  quantities  j>ik. 

As  we  have  seen,  the  ratios  of  y^,  art1  independent  of  the  partic- 
ular points  of  the  line  used  to  form  />,,..  If  in  particular  we  take 
one  point  as  the  point  0  :  ./'.,  :./•..:./•,  in  which  the  line  cuts  the  plane 
•'',  "  "•  'VVt'  ha\~c  /',.,—  —  •''..'/, ,  /'r.=  —.''.//,,  /',.  —  .r('/,  :  whence 
.r,:  .r.:  ./'4  - /',.,: /'I.T: /'u-  ^  s'",Ur  '"  ;i  similar  manner  the  points  in 
which  the  line  meets  the  other  coordinate  planes.  \vc  have,  as  the 
points  of  intersection  wit h  the  four  planes,  the  following  four  points  : 

0     :         /'.      :          )>...    : 


LINK  COOKDINATKS 

The  condition  that  these  four  jioints  should  lie  oil  a  straight 
lino  is  exact  Iv  the  relat  ion  (  I  ). 

From  (  •>  )  it  follows  that  a  set  of  ratios  /',,  can  belong  to  only 
one  line  and  that  these  ratios  niav  liave  anv  value  consistent 
with  (4). 

llelice  flu'  r<iti<>x  <'f  /',..   'until  Li'  /<//,-,//  r/.s1  ///••  fnilrilhi'ttfx  »(' <i  xti-iin/hf 

1<H'\    illlil    f/n'    /'I'/ilf/u/l    /,,///',,,/    ,1    Hfl'ttt'i//lf     lilli-    ilii'l    itfi    I'nfiri)  UHltt'H    IX    "III1 

f"  >mi'.    'I  hese  coordinates  are  called    /'///'<•/•»•/•  fin'lriJtnnti'x. 

()t  course  it  a  straight  line  lies  completely  in  one  ot  the  coor- 
dinate j i lanes,  one  ot  the  sets  of  ratios  in  (  •"> )  becomes  indeterminate. 
This  cannot  happen,  however,  for  more  than  t\\o  of  the  sets  at  the 
same  time,  and  the  other  two  sets,  together  with  (  \  ).  determine  j^,. 

128.  Dualistic  definition.  A  straight  line  may  lie  defined  bv  the 
intersection  ot  two  planes  //.  and  ?v,  lieasoiiinnp  as  in  £  }'2~  \\c  are 
led  to  place 


(1  ) 


\\'hich  are  connected  by  the  relation 


To  any  straight  line  corresjionds  one  ratio  set  of  ratios  ot  y,;. 
and  the  four  planes  through  the  straight  line  and  the  vertices  ot 
the  tetrahedron  of  reference  have  the  lane  coordinates 


Therefore,  to  any  set  of  values  of  the  six  quantities  </,  which 
satisfy  the  relation  (•_'),  there  corresponds  one  and  oiil\  one  line 
\\  it  h  t  he  coordinates  ,/  , . 

The  relation  between  the  quantities  />  .  and  y,.  is  simple.  I- r<>m 
('•'>)  I  he  plane 


304  FOIK    D1MKNSIONAL  (JKUMKTKV 

passes  through  the  line  7,,.    If  ./',  and  //x  are  two  points  on  the  line 
we  have,  besides  equation  (4),  the  equation 

7l-.-//-;+7l:>/'..+    7l4.'/4  =    °-  (-;>) 

Fn>m  (  \  )  and  (  ">  )  we  have 


Similarly,  wo  may  show  that 

7l2  _    71?  _    7l4   _    7:H   __    7.2  =    7ca. 
/':.4          /Y;          /'•-•«         /'.^         /'l3         /',, 

We  may,  accordingly,  use  only  one  set  of  (juantities 


and  may  interpret  in  point  or  plane  coordinates  at  pleasure. 

129.  Intersecting  lines.    Two  straight    linos,  one  determined,  by 

the  points  .r  and  j/i  and  the  other  by  the  points  j\  and  //|,  inter- 
sect when  the  four  points  lie  in  the  same  plane,  and  only  then. 
The  necessary  and  sul'ticient  condition  for  this  is 

r,       .r,      ./• ,       .r4  ,' 

.'/i      ,'/j      .'/:.      ,'/i      :0 

./•;    ./•;    ./-;    ./•; 

whi.'h   is  the  same  as 


Alsn.   dual isticallv,   twn   lines,   nne    dcMermined    by   the  planes   >it 
and    '•,  and   the   other  liv   the   planes   //   and    /'|,    intersect    when   the 


LINK  COORDINATES 


four   planes    pass   through    tin1   same    point,    and    only    thru.     Tl 
necessary  and  sufficient  condition  for  this  is 


which  is  the  same  as 


Either  condition  (1)  or  (-)  is  in  terms  of  /•,,, 


which  is  more  compactly  written  as 


where  M  (  >,  ;•'")  is  the  polar  of  the  quadratic  expression  <-/>(/•). 

130.  General  line  coordinates,  ('onsider  any  si\  <|iiantitirs  j-t  de- 
lined  as  linear  combinations  of  the  six  quantities  /-,,.  That  is,  let 

P-ri=  f*iiru+  '*i-Sr.;+  'r-:;''i.+  ",«'';.+  ">:,''*•>+  «,,<'^'  '  1  ) 

with  the  condition  that  the  determinant  of  the  coefficients  nit.\ 
does  not  vanish.  Then  the  relation  between  the  quantities  j>ti  and 
.r.  is  one-to-one,  and  j\  may  be  used  as  the  eoiu'dinates  ot  a  line. 

liv  the  substitution  (1)  the  fundamental  relation  <o(r)  -  ()  Ljoes 
into  a  quadratic  relation  of  the  form 

|(.;-)^:V,/i;.r.,-t.-  0.  (^..=  ,^)  (•-') 

In  fact,  bv  a  pro[)ei'  choice  ot  the  coefficients  in  (  I  ).  the  function 
£  (  ./•  )  may  be  any  quadratic  form  ot  nonviinishiutl  discriminant  and, 
in  [tarticular,  mav  lie  a  sum  ot  the  six  squares  ./••.  'I  lie  jtroot  ot 
this  mav  be  i^i\'en  as  a  ^enerali/.atiou  ot  the  similar  problem  in 
space  or  mav  be  found  in  treatises  on  algebra. 

P>v  the  substitution  (1  )  t  he  polar  r/>  (  /•,  /•'  )  ^oes   into  the  jmlar 


To  prove  this  let  rt>  and  /•',  reju'esent  t  \\  o  sets  of  values  of  the  coor- 
dinates /'^  and  let  .i\  and  ./•'  represent  the  corresponding  \alues  ot  the 
coordinates  .r.;  then  /•,,.  -f-  X/*',  corresponds  to  /^.-j-  \./'|  lor  all  values  oi  X. 

Therefi  ire  ro  (  /•  -f  X/-'  )—  g(.r  +  X.r'  ), 

or      co  {/')+  '2  \(D  (  /-,  /•'  )  -f  \'a>  (/•')—  f  (.r  )+  '2\%(  .1  .  ./•'  )   f    X'f  (  ./  •'  ). 


80fi  FOt"  K-DI  M  KNSION  A  L  (1K(  )M  KTHY 

By  equating  like  powers  of  \  we  have 

">('•.  >*')  =  £(•'•«  -'•')• 

Hence  the  r<ifi"x  "/  tin//  sj/steiH  <>f  si.r  tji/<intitii'n  ,r.,  hound  /<//  a 
Jionioi/fni'i'un  fjmidmh'c  relation  £(./•)=  0  ///'  nonvanixhiny  dinerimi- 
niinf,  ttitij/  l>e  ftikt'ii  tin  t/n'  ctiiirdinaten  'if  a  lint'  in  njitii-e  /;;  .sv/r'//  ^/ 
///iin  tier  t/nif  f//f  *'<{u<iti'>n  £  (  ./•,  ./•'  )  —  0  in  tin'  //fvr.v.\v//y/  and  sufficient 
rnntlition  for  ffif  intersection  of  tin'  tiro  linen  .r  (///</  ./•'. 

(  )t  particular  importance  are  coordinates  due  to  Klein,  to 
which  we  shall  refer  as  Klein  coordinates.  These  are  obtained  by 
the  substitution 


The  fundamental  relation  is  then 

•>T  +  .'\f  4-  .'V  +  ^J  +  .>•;  +  .'V  -  0, 
and  the  condition  for  the  intersection  of  two  lines  is 

•''i.vi  +  J'^'i  +  -'V/3  +  •'  4//i  +  -''.v'A-,  +  •'V,//--,  —  (  '• 

131.  Pencils  and  bundles  of  lines.  /.  //'  <t.  <m<l  ?>.  nrc  t/r<>  inter- 
nt'i-fin;/  1/tn'n.  tJten  p.r,  =  '/,  +  X/«.  /x  '/  ////*'  <//'  ^//c  f»'n<'il  dt'teriinnrd  />// 
n^  mill  l>r  itii'l  <i)iij  Jini'  at  the  pencil  >/>'i//  f>e  *<>  expressed. 

The  hypotheses  are 

£(Vr)=0,      f(A)=0,      |(/r,  /,)  =  «• 

Then: 

1.   .r,  are  the  coi'irdinatcs  of  a  straight    line,  since 

£<.'•)  =  ^(  "  4-  \r>  )  =  £  (")+  2  \£(  '/.  A  >  -f  x'J|(  /•  >  -  o. 

•J.  The  line  ./•.  lies  in  the  plane  of  ti:  and  lt  and  passes  through 
their  point  of  intersection.  To  prove  this  let  </  be  anv  line  cutting 

I  I  I  ».  t1! 

both  itt  and  /-  .  That  is.  <l.  is  either  a  line  through  the  intersection 
of  //i  and  A-  or  a  line  in  the  plane  of  ti.  and  1^.  Then  £(<t,  </)  =  (), 
and  f  (  A,  '/  )  =  o.  Therefore 


LINK  COORDINATES  :}l>7 

Hence  .i\  intersects  anv  an<l  nil  of  the  lines  </t  and  therefore  lies 
in  the  plane  of  <it  and  /<(  and  passes  through  their  intersection. 

•  \.  '1'he  value  of  X  may  l»e  so  taken  as  to  give  anv  line  of  the 
pencil  determined  Itv  <t:  and  /<,.  To  prove  this  let  /'  he  any  point 
of  the  pencil  except  its  vertex,  and  let  //,  he  a  line  through  1'  but 
not  in  the  plane  of  <it  and  1^.  We  can  determine  X  so  that 

£(./•.  /0  =  £(".  /'  >-f  xf  <'•*  //>=  o. 

Hence  .r,  intersects  /^  :  and  since  /^  has  oiilv  the  point  /'  in  the 
plain-  of  dt  and  /<f,  and  ./•_  lies  in  that  plane,  j\  passes  through  /'and 
is  anv  line  of  the  pencil.  The  theorem  is  completely  proved. 

//.  //'  */(,  /',.  it/nl  <•.  <!/•>'  th/'i'i'  lini'x  t/tfi'tii'/t  (/if  ml/tit'  j">int  f>nt  >/"t 
t't'luHt/t/li/  t"  thi'  8<IHH'  fit'Hi'tl,  t/tt'tl  p.t\  -----  ilt  -\-  \f't  -f-  /J.i\  /.">'  '/  //Hi'  tfti'"tii/h 
f/n1  mi/in'  j'l'int.  it/ni  iiiti/  lint'  t/i/'t>i(t//l  t/ulf  j/»mt  ///''//  If  .v/  >'<'j>/'t'xr/itt'<l, 

Uv  hypothecs,  f  (/O  =  0,  f  (A)  =  0,  f  ('•)  =  0,  |(</, /«)  =  0,  ^(/M')  =  0, 

£(  r,  ")=  (>-     Tlu-ii  : 

1.   ./•,  are  the  coordinates  ot  some  line,  since  ^(./')  =  (|. 

^.  Any  line  \\hich  cuts  all  three  lines  <r,  //..  and  jv  cuts  ./-(.  I'or, 
if  %(«.  c/)=0,  |(/.,  (7)=0,  and  |(<-,  t/)=0,  then  |(r.  ,/)=|(,n?) 

-f  X^  (/>.</)+  fj.%(  <\  ,J )  =.  0.     Therefore  ./•_    passes    tlirougli    the    inter- 
section ot    nt,  /'t.  f't. 

'•}.  N'alues  of  X  and  fj.  may  be  so  detcrmine(l  that  j\  may  cut 
an\'  two  lines  </t  and  //_  \\hich  do  not  cut  the  lines  </t.  /-_,  and  ct.  \\  e 
ha\  e,  in  fact,  to  determine  X  and  fj.  from  the  two  equations 

£<</,  //)  +  x^(/..  //)  +  M?C'-  //)  -  <>, 
£(.f,  //)+  Xf  (/',  /o-f  /"f  ('%  /<)=  0. 
'1  he  theorem  is  therefore  proved. 

///.  //  ",.  /',,  '//c/  »•.  /r/T  f//^/  tltrii'  liniK  in  tin'  x<i//ti-  j>l<(n>-  /'«(  //"f 
l'<'l"ii</< it*/  /"  tin'  xiintt'  jn'ni'il.  tin  H  p-t\-~~  '',  +-  XAt  +  /^'',  /^  ''  ////'  ///  ^/c' 
yitiHi'  fi/iUti',  ((HI/  iin/i  1 1 n>'  m  tin1  I'tit/ii-  uiiii/  In'  *"  r>  fi/'tfi'/t ti<l. 

The  proof  is  the  same  as   for  theorem    II. 

A  roiitigurutioii  consisting  ot  all  lines  tlirough  the  same  point 
is  called  ;i  I'H/iilli  of  lines.  A  eoiiiigunition  consisting  of  all  lines 
in  a  plain-  is  a  ///-///»•  of  lines.  l>v  the  use  of  line  eoJ'irdinates  \\  e 
do  not  distinguish  between  a  bundle  and  a  plane  of  lines.  In  tact 
each  coiiliguratioii  consists  of  a  doiihlv  infinite  set  ot  line--  each  ot 
\\  hieh  intersects  all  of  the  others. 


•JOS  FOUR-DIMENSIONAL  GEOMETRY 

EXERCISES 

1.  Trove  that  the  cross  ratio  of  the  four  points  in  which  a  straight 
line  meets  the  tour  planes  of  any  tetrahedron  is  equal  to  the  cross 
ratio  of  the  four  planes  through  the  line  and  the  vertices  of  the 
tetrahedron. 

'J.  Trove  that  there  are  t  \vo  and  only  two  lines  which  intersect  four 
L,rivcii  lines  in  general  position. 

3.  Trove   that    if  the   coordinates   of  anv    live   lines  satisfy  the   six 
equations  . 

*"''i   +    ML    +    »'-',    +    P'S',    +   <Tf,    =    ()' 

the  five  lines  intersect  each  of  two  fixed  lines. 

4.  Show   that    if  the  coordinates   of  anv    four  lines  satisfy  the  six 

equations 

A./-,  4-  it-!/,  +  v-t  +  px,  =  0, 

any  line  which  intersects  three  of  them  intersects  the  fourth,  and  hence 
the  lines  are  four  generators  of  a  quadric  surface. 

5.  Show  that  if  the  coordinates  of  three  lines  are  connected  by  the 

six  eiiuatioii> 

A./',  +  fj.i/1  +  w,  =  0, 

anv    line    which    intersects   two  of  them    intersects   the   third.     Thence 
deduce   that    the    lines   are   three   lines   of   a   pencil. 

132.  Complexes,  congruences,  series.  A  lute  c<»/i/>!e.r  is  a  three- 
dimensional  extent  of  lilies.  It  may  be,  lint  is  not  necessarily, 
defined  by  a  single  equation  which  is  satisfied  by  the  coordinates 
of  the  lines  of  the  complex.  The  <>/•'/>'/•  of  a  complex  is  the  num- 
ber of  its  lines  which  lie  in  an  arbitrary  plane  and  pass  through 
an  arbitrary  point  of  the  plane:  that  is,  it  is  the  number  of  the 
lines  of  the  complex  which  belong  to  an  arbitrary  pencil. 

A  lini'  I'nii'jnii'iii'i'  is  a  two-dimensional  extent  of  lines.  It  may 
lie  defined  by  two  simultaneous  equations  in  line  coordinates  and 
is  then  composed  of  lines  common  to  two  complexes.  The  "/•</<•/• 
of  a  congruence  is  the  number  of  its  lines  which  pass  through  an 
arbitrary  point  :  its  /-A/xx  is  the  number  of  its  lines  which  lie  in  an 
arbitrary  plane. 

A  line  writ-*  is  a  one-dimensional  extent  of  lines.  It  may  be 
defined  by  three  simultaneous  equations  in  line  coordinates.  It 
then  coiisi>ts  of  lines  common  to  three  complexes.  The  "/-<Av  of  a 
scries  is  the  number  of  its  lines  which  intersect  an  arbitrary  line. 


LINK  COORDINATES 

An  equation  .^'(•/'1<  •''.,•  •''.•  •''.,•  -'V  •'',•)—  "< 

where  /'  is  a  homogeneous  polvnomiul  of  tin-  /ah  decree  in  j  , 
defines  a  line  complex  of  tin-  //th  order.  Let  << t  and  f>t  lie  an\  two 
lixed  intersecting  lines.  Then  <it  -\-  >/-,  is,  bv  thei  >rein  I,  s;  1  •'>  1 ,  a  line 
of  the  pencil  defined  hv  <ti  and  f>t,  and  this  line  will  belong  to  the 
complex  (  1  )  when  X  satisties  the  equation 

/(  -/!+  \f>r  tf.-f  X/<.,.  f/..-f-  X/-.,  <^-f  X/<4.  (/.+  X/'..  </t.  -(-  X/v  )=  0, 

which  is  of  the  //th  decree  in   X. 

From  the  ahove  it  follows  that  through  anv  lixt-d  point  of  space 
^oes  a  conti^ui'at  ion  of  lines  such  that  n  of  these  lines  lie  in  each 
plane  through  the  lixed  point.  Since  the  relation  l>et\veen  the 
coordinates  of  the  fixed  point  and  those  of  anv  point  on  a  line 
of  the  complex  is  an  analytic  one,  derived  from  (  1  ),  it  follows 
that  <tn>/  ji'nnt  <>f  */»/'-,'  As-  tin'  t'ffti.r  <>f  it  i'<>tn'  «/  nth  -//•,/,•/•  fnrin<-<{ 
I'll  liitf*  '//  tin'  C"IH i'!f.r. 

Also  if  we  consider  a  lixed  plane,  through  e\er\  point  of  it  e;o 
//  lines  of  the  complex.  Since,  as  before,  we  have  to  do  with  an 
analvtic  equation,  we  infer  that  in  nni/  j>/<(/tf  t//>-  Hm-a  -./'  -r  c»m^L.r 
t'/ti'i'/i.'ji  <t  i-u /•>'>•  i >f  tin-  ntlt  i-litx*. 

A  simple  example  of  a  line  complex  is  that  which  is  composed 
of  all  lines  which  intersect  a  lixed  line.  For  if  ./  are  the  coordi- 
nates of  a  fixed  line  .1,  the  condition  that  a  line  r  should  intersect 

A  is,  liv  $  l:>o, 

-    s  £  ( •/.  ./•)  =  <»,  <  _  ) 

which  is  a  linear  equation.  Hence  this  complex  is  of  the  first 
order.  In  fact  through  an  arbhrarv  point  in  an  arlutrarv  plane 
iroes  obviously  only  one  line  intersect  in^  .  I.  Throiiu'li  a  fixed  point 
.!/  >_roe>  a  pencil  of  liiie>:  nanieh,  the  lines  through  .17  in  the  plane 
determined  liv  M  and  .1.  '1'his  is  a  cone  of  the  !ir.-t  ordei'.  In  an\ 
plain-  //(  u'oes  a  pencil  of  lines;  namcK,  the  lines  through  the  point 
in  \\liich  ///  interx-cts  ./.  The>e  form  a  line  extent  ol  the  fust  cla.-s. 
Another  example  of  a  line  complex  is  one  of  second  order 
defined  bv  the  eijualion 

/'i.  +  //;,  +  /'n-f-/'-,  +  /'^  +  /'-j  -  (l-  (:{) 

wliich,  expressed  in   point   coordinates,   is 

(./•,//—    ./    ,//i     )-    4      (.',//  .'      //.    )'"     f    <•',//!  -',.'il]        '      '    •'      .(/,  •'   ;•'       '" 

-f    (    ./      //  ./'    //       I'     \-    (    .1      'I  .1      <l       )"'--     H.  (      1     ) 


310  FOUR-DIMENSIONAL  GEOMETRY 

This  is  not  the  equation  of  a  surface,  since  it  contains  two  sets 
of  point  coordinates.  It",  however,  the  coordinates  //,  are  fixed, 
(4)  becomes  the  point  equation  ot  the  cone  of  second  order  formed 
bv  lines  of  the  complex  through  //,. 

If,  dualisticallv,  we  express  equation  ( •'» )  in  plane  coordinates  », 
and  i\  and  hold  rt  fixed,  we  obtain  a  plane  extent  of  second  class 
in  M(  which  is  intersected  by  the  plane  r_=  const,  in  a  line  extent 
enveloping  a  curve  of  second  class. 

Through  an  arbitrary  point  in  an  arbitrary  plane  L,ro  two  lines 
of  the  complex  ( •> ). 

An  example  of  a  line  congruence  is  that  of  lines  intersecting 
two  iixed  lines.  It  is  represented  by  two  simultaneous  equations 
similar  to  (-).  It  is  ot  the  lirst  order,  since  through  anv  point 
but  one  line  can  be  passed  intersecting  the  two  Iixed  lines.  It  is 
of  second  class,  since  in  a  fixed  plane  only  one  line  can  be  drawn 
intersecting  the  two  fixed  lines. 

Another  example  of  a  line  congruence  consists  of  all  lines  through 
a  point.  This  is  of  first  order  and  /.ero  class.  Still  another  example 
consists  of  all  lines  in  a  plane.  This  is  of  /.ero  order  and  first  class. 

An  example  ot  a  line  series  is  thai  of  lines  which  intersect  three 
fixed  lines  and  is  represented  by  three  linear  equations  of  the 
form  (-).  Such  lines  are  one  family  of  generators  on  a  surface  of 
second  order  (§  9tJ).  The  series  is  of  second  order,  since  anv  line 
in  space  meets  two  lines  of  the  series. 

133.  The  linear  line  complex.    The  equation 

a f/'i  -f-  ''...''.,  +  "..-'' ,  +  <i ,./',  +  '<..''.  -f-  nt.j-f  =  0.  (  1  ) 

where  ./•,  are  general  line  coordinates,  defines  a  linear  line  complex. 
An  example  of  such  a  complex  is,  as  we  have  seen,  that  which  is 
(•(imposed  of  lilies  cutting  a  fixed  line.  Such  a  complex  we  call  a 
xfH'i-ittl  linfiir  Inn-  I'mnjilfjr  <  >r,  more  concisely,  simply  a  .sy«-<v,//  <-«//<y /»•./•. 
'fhe  nece.-sarv  and  suilicient  condition  that  (1  )  should  represent  a 
special  complex  is  that  the  equation  (  1  )  should  be  equivalent  to 

£<  •'-.  // ,  =  0  ; 

that  is,  that  pnt  -  (  'J  ) 


LINK  COORDINATES  ;)  1 1 

Equations  ('2)  can  he  solved  for  //,  since  the  discriminant  of  ( •'> ) 
does  not  vanish  (  sj  1  :><>).  The  results  of  the  solution  substituted 
in  (  ;>>  )  n'ive  a  relation  <>t  the  lorm 

where  i/(n)  is  a  homogeneous  tjuadratic  polynomial  in  <it. 
\Ve  sum  up  as  follows  : 

7.  ,  1  xiH'cinl  ItlH'tti'  t'<i>/ijt/i'.i'  /x  1'innimxt'd  <//'  ittrat</Jit  li/tt'x  //'///'•// 
iiifi'/'fti'i'l  i>  fl./'f'i/  ///ii1  I'ltlli'il  trtf  (ij'ix  at  tin'  cnmiilfJ',  A  li/ii'ii/'  ii/K'i- 
fi'i/i  (  1  )  i/i'fitti'*  ii  Nfn'ci'ttl  i'"in[>l,'.r  U'lu'ii  <tml  "iilij  u'/n'n  flu-  roi-jficirnts 
dt  xdtixt//  tin'  (j)nn?t'(ttia  1'ijimtiun  (4). 

.More  in  detail,  let 
Then  ('([nations  (  '2  )  are 

from  which,  together  with  (•')),  we  have 

n  it  +".,//.,  +  <i.. //..+  <*//  +  (i.  n.  -f-  "  . '/.  —  "•  (7) 

l'"rom  ( ii  )  and  (  7  )  we  obtain 

«, 
(i , 


.VI  ;,  t 


where  JiA  is  the  c-ofaetor  of  (/ijt.  in   the  expan>ion  of   />  —    </ ^  '. 


Then  =Anni  +  .l  .,„.,+  , I  ,,r,  +  .1 

en 

/> 

>/ . 
P 


312  tWK-blMENSlOXAL  GEOMETRY 

We  may  sum  this  up  in  the  following  theorem: 

//.    Tlii'  (•(x'irdinafcx  'if  tin1  a.rix  <>f  the  cmnple.r  (  1  )  irhcn  it  in  xpecial 

are        •    If  Klein  coordinates  <tr<'  iixea1,  the  c'H'ira'inatex  /if  the  a.i~ix  <>f  <i 
f<ti 

xpe''/iil  en//ip/e.r  /t/'i'  tlif  coefficients  m  flic  eipiatioii  />f  tin'  complex. 

Returning  to  the  general  linear  complex  (1)  (special  or  non- 
special),  consider  any  point  /'.  It'  at,  />,,  and  <-t  are  any  three  lines 
through  /'  not  in  the  same  plane,  then  (theorem  II,  £  lol)  any 
line  through  /'  has  coordinates  ",-H  X/^-f-  /j.e,  and  this  line  belongs 
to  the  complex  when 


Kijtiation  (X)  is  satisfied  for  all  values  of  X  and  fj.  if  the  three 
lines  <it,  //.,  and  r(  l)clono-  to  the  eomplex.  Otherwise,  assuming 
that  f.  does  not  belong  to  the  complex,  we  may  solve  (  S)  for  /z 
and  write  the  coi'n-dinates  of  the  point  ,<,';  in  the  form 


where  (f'  and  !>[  are  two  delinitely  defined  lines  through  /',  and  X 
is  arbitrary.    This  proves  the  following  theorem: 

///.  Through  (in//  arbitrary  i»nnt  tn  XJHIO'  </i>cx  <i  pencil  <>f  linen  »f 
th>'  i''i///i>/('./'  unlt'KX  in  an  exceptional  imonoT  (til  line*  thruwjli  tin'  jminf 
bt'lnwj  tn  t)u'  cn/npli'.r. 

Tlie  analysis  would  be  the  same  if  the  three  lines  <r,  !>.,  and  <•. 
\vere  taken  as  three  lines  in  a  itlane,  but  not  through  the  same 

I  O 

point   (theorem   III,   ^lol).     Hence 

IV.  In   iini/  arbitrary  plane  in  ,vy»/<r  I/ex  <i  pencil  t>f  liiifx  <>f  ///«• 
rnniph'j'  littlest  tn   an   exceptional  manner  aU  linex  "f  f/n-  plane  l»i»n<j 
t"  t/ie  eo/nplej'. 

To  complete  the  information  given  bv  these  two  theorems  we 
>hall  prove  the  two  following: 

V.  If  ill!    f/'nex    tlu-'illijll    dill/    n/ii'    paint    /'   ae/a,i,/    fu     tin1    f"  I/I  /'li'.r,    t/te 
enniplt'j-   ix  sperm!  miJ  tli>'  point   I'  I/ex  mi   the   a.rix  <f  the   e<>///p/e.r. 

Let  all  lines  through  /'  (l''ig.  :'>*>)  lie  lines  of  the  complex.  Take 
A,  a  hue  in*!  belonging  to  the  complex,  ;md  let  (t>  ami  A'  be  two 


LINE  COORDINATES 


313 


points  of  It.     Through  ({>  goes,  by  theorem  III,  a  pencil  of  lines  of 

the  complex  of  which  l'(t>  is  evidently  one  and  //  is  not.    Similarly, 

through    A'  goes  a   pencil    of   lines    of   the   complex   of   which   A'/'   is 

one  and  //   is  not.     These  two  pencils  lie  in  different   planes,  for  if 

they  lay  in  the  same  plane  the  line 

//    would    lie    in    both    pencils    and 

be  a   line   ol    the   complex,  contrary 

to    hypothesis.    The    planes    of   the 

pencils    intersect     in    a    line    which 

contains  /'.    Call  it  c,  and  let  S  be 

anv  point  on  <•. 

The  line  SI'  belongs  to  the  com- 
plex, since,  bv  hypothesis,  all  lines 
through  /'  are  lines  of  the  complex. 
The  line  .S'((>  belongs  to  the  com- 
plex, since  it  lies  in  the  plane  of  the  pencil  with  the  vertex  (,>  and 
passes  through  (t>.  Similarly,  the  line  A'A'  belongs  to  the  complex. 

Therefore  we  have,  through*  the  point  .S',  three  lines  of  the 
complex  which  are  not  eoplanar,  since  c  and  It  are  not  in  the 
same  plane.  Hence,  bv  theorem  III,  all  lines  through  S  belong  to 
the  complex.  Hut  >>'  is  anv  point 
intersect  '•  form  a  complex,  the 


theorem  is  proved. 

VI.  If  nil  HiK'x  <>f  «  j>l<ut>'  /<«'- 
/"////  /'/  the  CiDHplt'JC,  till'  COIHplt'J' 
is  XJK't'Utl  'lltd  tin-  pliliti'  IKIXXI'X 
through  tin'  a.iix  of  the  n>nt/il>'.r. 

Let  all  lines  of  a  plain-  in 
(Fig.  -r>7)  belong  to  the  com- 
plex. Take  //,  anv  line  not  of 
t  he  ci  uii[ilcx,  ami  let  '/  and  /•  lie 
two  planes  through  //,  intersect- 
in ///  in  the  lines  m  and  inr. 


>f  c,  ami  since  all  lines  which 


Kit:.  .".7 

1  n  the  plane  Y  lies.  b\  theorem  1  \  . 
a  pencil  of  lines  of  the  complex  ol  which  //<y  is  one  and  h  is  not. 
Similarly,  in  the  plane  /•  lies  a  pencil  of  lines  of  the  complex 
ot  which  ////•  is  one  and  //  is  not.  These  pencils  have  different 
veriice-s,  for  otherwise  they  would  contain  h.  Let  <•  be  the  line 


314  FOUR-DIMENSIONAL  (JKOMETKY 

connecting  tin-  vertices  (  -•,  of  course,  lies  in  >/t  ).  Take  a,  any  plane 
through  '•  intersecting  '/  in  the  line  y*  and  r  in  the  line  rx. 

Then  c  is  a  line  of  the  complex,  since  by  hypothesis  any  line  in 
///  belongs  to  the  complex.  Also  yx  and  rx  belong  to  the  complex, 
since  each  is  a  line  of  a  pencil  which  has  been  shown  to  be  com- 
posed of  lines  of  the  complex.  The  three  lines  do  not  pass  through 
the  same  point  because  <////  and  nn  have  been  shown  to  intersect  <• 
in  different  points. 

Therefore,  bv  theorem  I\'.  all  lines  in  .s-  belong  to  the  complex, 
and  since  x  was  any  plane  through  '•,  all  lines  which  intersect  c 
belong  to  the  complex,  and  the  theorem  is  proved. 

134.  Conjugate  lines.  Two  lines  are  said  to  be  foiijut/atc,  or  /v- 
i-'>l>r<i<-iil  }i<>l<irx,  with  respect  to  a  line  complex  when  every  line  of 
the  complex  which  intersects  one  of  the  two  lines  intersects  the 
other  also.  Let  the  equation  of  the  complex  in  Klein  coordinates  be 


and  let  //,  and  zt  be   the   coordinates  of  any   two   lines.    The  condi- 
tions that  a  line  j;  intersect  //,  and  zi  are  respectively 

yj  +  a-   +  -v'  +  'J  +  //-'  +  .     =  '  *'  (  -  ^ 


\Ve  seek  the  condition   that   any  line  .r,  which  satisfies  (1)  and  ('2) 
will  satisfy  (  :!  ).    This  condition  is  that  a  quantity  X  shall  be  found 

such  that 

pz—yt+Xit;.          (t  =1,  '2,  :>,  4,  ;>,  »•)  (4) 

lint  //,  and  ^_  both  satisfy  the  fundamental  relation 


1  herctore,  from  (4),       X=       _^—  "J,  (  f, 

V<r 

•*-v    ' 

-2'VA 

and  (4)  becomes  pz  —  i/-  '-'  (/.,  (  »; 


which  define  the  coordinates  .r  of  the  conjugate  line  of  any  line  >/,- 
l'"i'om  (.))  follows  at    once  the  theorem: 

/.   .I/'//   ///*••   //'/N  '/    ti/H'/Ui'   <•">(/  i/i/iif'' 

i  'Hill  Ji  /f./1. 


LINK  ('OORDINATKS  3K, 

If  the  lino  //,-  l>elongs  to  the  complex,  then  ^",//,  =  "  and  pz,=  '/,- 
Hence 

//.    Am/    Illti'    i>1    ll    HilUHflt't'lH/    I'minili'X    IK    itx    nll'il     I''/// />/,//!  f  i'. 

If  the  complex  is  special,  N^/j!=0.  Therefore,  unless  also 
"V 'f  ,-//,=  0,  X—  x  anil  pz,=  ii,.  Ilence 

///.  77/i'  '/./•/*  "/'  (/  vjn'i"/'<i/  cvntplfj'  />•  ///<•  fiinj'ttifitff'  >>('  <>/i//  I'm,'  ii"t 
Iii'l'iti'l'mi/  f"  th>'  >•<>,, //>/i'.r. 

If  the  complex  is  special  and  the  line  //,  belongs  to  it,  A.  is 
indeterminate.  I  Icnce 

7V.  .1   (tin'  i if  it  xjit'i'iitl  i-'iiii  fiji-.r  Ji'ix  n»  ifi-f,  ruii/i/it-'  <'»nj m/nti'. 

The  aliove  theorems  mav  also  lie  proved  easilv  hv  purely  geo- 
metric methods. 

If  two  lines  have  coordinates  //1  and  ,r,  u'hich  satisfy  t^j nations  ( ti), 
then  any  values  of  ./•.  which  satisfy  (  i' )  and  (.'))will  also  satisfy  (1). 
I  Icnce 

V.  If  f"'"   ff>i»'x    iifi'    ctiHJHi/titt'    iritli    rrspi'i-t    In   ,i    <•<  1 1  it  j  i]  ,-.r,    inn/   Inn 
H'Jiii'h    i'tift'rfii'1'fit  fii'f/i   ' >f  tJtt'i/i   ?H*ftiHf/x  f"  tin'  <'«ni i>li'.r. 

I-'rom  this  theorem  or  from  the  relations  (  (I  )  t'ollnus  at   once: 

VI.  Tn'n  luii'x  fu// / iii/iifi1  //•///!   /•, '.ijit'i-f  fu  ft  ttunxpi'i'idl  i'<>/ii [ilt'.i  iln  >i"f 
inft'rxi'rf. 

\\'e  have  seen  (theorem  IV.  ^  loo)  that  in  anv  plane  ///  there  is 
a  iinit|iie  point  /'  which  is  the  vertex  of  the  pencil  of  complex 
lines  in  />/.  Similarly,  through  anv  point  /'  '_nies  a  plane  ///  which 
contains  the  pencil  ot  complex  lines  through  /'.  \\  hen  a  point  and 
plane  are  so  related,  the  point  is  called  the  //«/••  of  the  plane, 
and  the  plane  is  called  the  j>ul,ir  of  the  point. 

It'//  and  //  are  two  conjugate  lines  with  respect  to  a  complex, 
and  /'  is  anv  point  on  //.  the  pencil  of  lines  from  /'  to  poinis 
mi  //  is  made  up  of  complex  lines  l>v  theorem  \  .  Ilence  tollo\v 
the  theorems: 

vii.  'rii>'  /"J,ir  fihtuc  i  if  ,i  /,.,;///  /'  ,,,,  ,i  /;//,•  _,/  /x  ///,  j.inn-  ,1.1,1-- 

itii it'll    I'll    /'    ilifl    tit,'     i-niijlli/illt      till,     It.       .l.v     /'    l/l'il'rX    itli'llij    '!    t)t>     fin/iir 

Illll/ll        tll/'l/S      llln.llt      //. 

VIII.  Tli''  /'"/<•  ni'  int'i  ill, n/,'  n>  ////•"//'///  ii  />'//'  '/  /'>•  ///••  htfi /'xfi'ft'i'ii  "'' 
in  ii'ith  ///<•  i'un  i  ui/<it  i'  ///!>•  h.  A*  in  fa  nix  iiliniit  ,/  it*  ji>J,'  frifi-fxi's  Ii. 


olO  FOLK    DIMENSIONAL  GEOMETRY 

135.  Complexes  in  point  coordinates.    It  is  interesting  and  instruc- 
tive to  consider  the  linear  complex  with  the  use  of  point  coordinates. 
A  linear  equation  in  general  line  coordinates 

^".ri  —  (1) 

is  equivalent  to  a  linear  equation 


in   /',,.  coordinates,  and   this,  a^ain,  can   be  expressed  as  a  bilinear 
equation  in  point  coordinates: 


If  in  equation  (o)  we  place  //,  equal  to  constants,  the  equation 
becomes  that   of  a  plane  ni  of  which  //_  is  the  pole. 
The  plane  coordinates  of  this  plane  are 

P"l=  'V'2  +   V/,+   ",,/V 

P"8=  -",=//!  +  <Vs-  V/4'  4 

P";.=  -"13.'/i-'V'2  +  'W 

P^=-  VVi+'V/2-'V/3< 

and  to  each  point  //,  corresponds  a  unique  plane  unless 


-  ",a  "M      -  ^ 

—  it        —  a  0  <  i 

i  .1  •:  i 

-a  a  a  0 

11  fj  .i 


that  is,  unless  (  '',.,''.,-(-  "1;",.,  +  "i,''-,  ^"—  "• 

I)iit  '/  /'  4-  '/  /"',.,  4-  "U".V1  's  'n(>  form  which  ?/  (  't  >  takes  for  the  p  l: 
coi'u'dinates.  Hence  we  have  a  verification  of  the  fact  that  in  a 
nonspecial  complex  anv  plane  has  a  unique  pole. 

Let  us  take  two  conjugate  lines  as  the  ed^es  All  (./•=<">,  .r,=  0), 
and  <  '  I  >  (./•.  (•.  ./•  n  )  of  t  he  t  ct  rahedroii  of  reference  fur  t  he  point 
coordinates.  I  Ins  can  alwavs  lie  done  bv  a  collincation  \\'lm-h 
obviously  amounts  tn  a  linear  sul»stitut  ion  of  the  line  coordinates. 

If  0  :"://:  //    is  a  point    /'on   All.  its  polar  plane  is,  bv  ('•)), 


LINK  mulIDINATKS  317 

This   plane   must    pass   throii'di    <'!>  for  all  values  of   >/    and    >/ 

•  \~ 

Hence  a.^=ct,  =  a    —  <t  .    =  0,  and  the  line  complex  reiluce.s  to 


where     neither     ot     the     enettieiellts     call     In-     /ero     if     the     complex     i 

nonspecial. 

It  is  possible  to  make  the  ratio  <',,:  </     equal  tn       1   l>v  a  colline 
ation  of  >pace.     To  see  tliN.  note  that   it   we  place 


th 


Consider  no\v  a  special  complex,  and  let  its  axis  lie  taken  as 
the  line  .1/1  (  j-—  0,  ./•„=<>).  the  line  eoiirdinates  of  \\hidi  are 
y.,._,  =  y,.,=  /'14=  /'4.=  /'-.•.  =  "•  'I"ne  <>()i)dition  that  a  line  >houid  inter- 
sect this  line  is,  h  (!),>{  1  -'.», 


\Ve  mav  sum  up  in   the  following  theorem 


136.  Complexes  in  Cartesian  coordinates.  We  >hall  now  considt  r 
the  properties  and  equations  nf  Hue  complexes  \\ith  the  u>e  "t 
('artesian  coordinates  .r  :  //  :  2  :  /.  1>\'  which  the  plane  at  intinity  is 
unique  and  metrical  properties  come  into  evidence. 

For  special  complexes  we  have  two  cases,  according  as  the  axis 
i<  or  is  not  at  inlinitv.  In  the  former  case  the  linev  \\-hich  inter- 
sect it  ai'e  parallel  to  a  tixed  plane,  llem-e 


318  FOUR   DIMENSIONAL  (iEOMETRY 

Consider  a  nonspecial  complex.  In  the  plane  at  inlinitv  is  a 
unique  point  I,  the  pole  ot  tin1  plane.  The  lines  of  space  which 
pass  through  I  form  a  set  of  parallel  lines  not  belonging  to  the 
complex.  These  are  called  the  diameters  of  the  complex.  Each 
diameter  is  conjugate  to  a  line  at  infinity,  since  the  conjugate  to  a 
diameter  must  meet  all  the  pencil  of  lines  of  the  complex  whose 
vertex  is  I.  Conversely,  any  line  at  inlinitv  not  through  I  has  a 
diameter  as  its  conjugate.  In  other  words,  thr  polar  plane*  of  point* 
ot)  a  diameter  are  parallel  planes,  and  the  poles  oj  an  if  peneil  <if  paral- 
lel plant-*  lie  on  a  diameter. 

Consider  now  the  pencil  of  parallel  planes  formed  bv  planes 
which  are  perpendicular  to  the  diameters.  Their  poles  lie  in  a 
diameter  which  is  unique.  Therefore  there  i*  in  eai'h  non*/>ecial 
i'o)/tple.r  a  uniiiue  diameter,  called  the  a.ri*,  f/'hich  ha*  the  properf//  of 
fieini/  perpendicular  to  the  polar  plane*  of  all  point*  in  if. 

Referring  to  (4),  £  !•>;>,  if  we  replace  r:.ro:.r:.r  bv  .r://:z:f, 
the  pole  of  the  plane  at  infinity  is  given  bv  the  equations 

a    i/  4-  a    z  +  a    f  —  0, 

!-•'  l:l  14 

—  a    .r  -f-  a    2  —  a    f  —  0 

12  23  •!•_' 

-  a   ./•  —  a    ii  +  a    f  =  0, 

in  23«'  .:( 

which  have  the  solution 

./•://:  z:  f  =  «,.,:  -  ar,:  a^:  0.  (1  ) 

Any  line  through  the  point  (1)  is  therefore  a  diameter,  and  if 
( ./• ,  //p  z  )  is  any  finite  point  of  space,  the  equation  of  the  diameter 

through  it  is 

./•       .r,  =  //-//,  =  z-~\ 

a  —  a  a 

•2'.',  is  12 

The  polar  plane  of  ( .rr  //,,  z^)   is.  by   (  I  ),  £  1  :)-">, 


The  line  (1)  is  perpendicular  to  the  plane  (  L!  > 


Consequently,  if  (J\,  //f  ^  )  in  ('.})  ai'e  i-ejdaced  liy  xai'ialdc 
roiinlinates  (  j\  //,  z  ).  e<|uation  (  •>  )  becomes  the  ('artesian  et/nafion 
of  ///*'  '/.r/x  of  the  complex. 


LINK  COORDINATES  lilt) 

Let  us  take  this  axis  us  the  axis  of  z.  Thru,  from  (1),  ".,,-<>. 
a  —  0,  and,  from  (:>),  since  the  origin  of  coordinates  is  on  the  axis, 
<j  =  0,  (t  =  ().  The  equation  of  the  complex  is  then 

"»/'«+  "*/'*=°'  <•*> 

which  agrees  with  (•">).  ^j  1  ;>~>. 

In  ('artesian  coordinates  equation  (  -t  )  is 

,/•//'          ./•'//  -f   /,-(•  Z     )-=<»,  (;')) 

which  associates  to  any  point   (./•',  //'.  .:'  )  its  polar  plane. 

From  (  .")  )  it  is  ohvious  that  the  polar  plane  of  /'(./'',  //',  z'  ) 
contains  the  line  ./•//'  --./•'_//  —  (>,  2  =  2',  which  is  the  line  through 
/'  perpendicular  to  the  axis.  The  normal  to  the  plane  makes  with 


where    /•    is    the    distance    from    /'   to   the    axis.     This    leads    to    the 
following  result  : 

Tlii'  [xil'tr  pi,  nn'  /if  unif  IK,  hit  /'  <''>nfiiinH  tli<>  I'm,'  f/<r<>ni//>  /' 
nei'pftuUculttr  t"  flit'  ttj'tx.  If  I'  ix  mi  fin'  <U'>K,  ?/x  i>n]nr  jilnnf  ix  j>>r- 
nt  >/<//<'ii/<ir  tu  tin1  i/.rfx.  Ax  /'  ri'i'i'il.t'x  frmii  t/»-  <i.r/x  <i/n/i>/  <t  //'//, 
ni>riii'>i<UcuIi()"  f"  it,  tJn'  ii'in/in]  jiliiin'  tiinix  tifm)if  thix  pi'rni'ndii'ul<(i', 
tli,'  i/u'i'cfi'>n  iitn/  iiuKiinit  nf  r'itiiti"ii  ili'/n  ii'liiti/  njmn  flu'  x/<//>  mi<l  ///•• 
fit/Hi'  <>f  Ic.  If  I'  n/oi'cK  ///»n</  ,t  //'//,'  jxit'iit/,'/  /"  ///»•  <t.rfv,  ifx  i>i,1,tr 

jilillli'     IH'it'i  X     n,l/-illl'l    t"     //*'//. 

Anv  line  of  a  complex  may  lie  defined  liy  a  point  (  ./•.  >/.  ,~  )  and 
its  neighboring  point  (./•  +  </.>•,//  f  '///.  :•  \-<h).  If  in  (•>)  we  place 
./•'  —  j-  -\-  i/.i\  if'  1=  if  -f-  <///,  z  --  z  +  '/r,  we  have 

./•'///  —  i/il.r  --  Idh     •  0,  (  ti  ) 

which   mav  he  called   the  i///r.-/v////W  ^.^/.iti^it   <<>'///,'  fnwfift-.r. 

Filiation  (it)  is  of  the  tvpe  called  noninte^rahle,  in  the  sense 
1  ha  I  no  solution  of  t  he  form  /  (  ./  .  v.  -'-  (J  )  ~  "  can  1  »»>  found  tor  it  . 
It  is  satisfied,  however,  in  the  first  place,  l>v  straight  lines  who>e 

eiiuations  arc 

Z  —  f,  //     -  in.r.  (  (   ) 

In   the  second   place,   on   anv  c\lindei-  with   the  equation 

./•'  4-  >r    •  <r  (  S  ) 

mav  he   found   curves  who<e  direction  at   anv  point    satieties  (  (i  >. 


320  FofR  -DIMENSIONAL  CJEOMETKY 

For  the  direction  of  any  curve  on  (S)  satisties  the  equation 

.r</.r  -f //•///  =  0, 
and  this  equation  combined  with  ( i> )  <^ives  tin1  solution 

(^ 


'2  TTlf 

which  are  the  equations  of  helixes  with  the  pitch  "- 

k 

It  appears  from  the  preceding  that  any  tangent,  line  to  a  helix 
of  the  form  (  '.'  )  is  a  straight  line  of  the  complex.  \Ve  shall  now 
prove,  conversely,  that  any  line  of  the  complex,  excepting  only 
the  lines  (7),  is  tangent  to  such  a  helix. 

Since  z  is  assumed  not  to  he  constant,  we  may  take  the  equation 
of  any  line  not  in  the  form  (7)  as 


with  the  condition  In  —  }»n  —  k,  which   is  necessary  and   sufficient 
in  order  that  equations  (1«)  should  satisfy  (»>). 

The  distance  of  a  point  (  ./'r  //r  zl  )  on  (1»)  from  n/,  is 


.  in"  +  n~  )  zf  +  -2  (  >nl>  -f  n/>  )  ^  +  /--'  -f  /'2. 

It   is  easily  computed  that   this  distance  is  a  minimum  when 
////<  4-  >'/' 


i 
The  minimum  distance  is  ,  which  we  shall  take  as  n  in 

N    )li~-\-   >l~ 

the  equations  of  the  helix   ('.*)•     The  direction  of  the  helix   at  the 
point   (.7-j,  //r  -=1  )  is 

'/./•  :  <///  :  <lz  —  ~  i/i  :  .1^  :       —  m  :  »  :  ]  . 

\  his  is  the  direction  of  t  lie  line  (  1  "  ).  and  our  proposition  is  proved. 
\\  e  have,  therefore,  the  following  theorem: 


LINK  COORDINATES  ;;-J  1 

137.  The  bilinear  equation   in   point   coordinates.    The    equation 

^  ",*•'•,//<— (J  <  1  , 

is  the  most  general  equation  which  is  linear  in  each  of  the  two 
sets  of  point  coordinates  ( ./^ :./•„:./•.,:  r(  )  and  (//,://://  : //  ). 

IJy  means  of  (1)  a  definite  plane  is  associated  to  each  point 
//,.  its  equation  being  obtained  by  holding  >i:  constant  in  (  1  j. 
Similarly,  to  each  point  .r  is  associated  a  definite  plane. 

In  this  book  we  have  met  two  important  examples  of  equation  (1  ). 

I-  "k, =  ''.<•  Equation  (1)  then  associates  to  each  point  v,  it-- 
polar  plane  with  respect  to  the  quadrie  surface 

^",,^•,  =  0. 

The  pole  does  not  in  general  lie  in  its  polar  plane.  Exceptions 
occur  only  when  the  pole  is  on  the  qnadrie. 

II.  ati—  —  dfri  whence  ^i(=(l.  Equation  (1)  associates  to  each 
point  //.  its  polar  [)lane  with  respect  to  the  line  complex 

2<w*=o. 

The  point  //,  always  lies  in  its  polar  plane.  This  association 
of  point  and  plane  forms  a  null  *j/*fi'>ti,  mentioned  in  ^  10:2,  and  here 
connected  with  the  line  complex. 

EXERCISES 


3.  Prove  t  liat  a  complex  is  determined  hv  1  \vn  pnnx  ot  conjugate  lines 

4.  Prove   that    if   a    line   descrihes   a    plane    pencil    its   conjiiLrate    ;ds 
descrihcs  a   plane   pencil,  and   if  a  line  describes  a  qnadrie   surface   K- 
con  jugate  does  al>o. 

5.  Prove  tliat  a  complex  ('or  null  svstent )  is  in  general  delerniined  h 
any  three  points  and  t  heir  ]>«>lar  planes. 

G.    Prove   that   anv  two   pairs   of   |io!ai'  lines  lie   mi   tin-   same  quadri 
surface. 


322  FOUR-niMEXSIOXAL  (JKOMKTKY 

138.  The  linear  line  congruence.    Two  simultaneous  linear  equa- 
tions in  line  coordinates, 

define  a  congruence.  Evidently  equations  (1)  are  satisfied  by  all 
lines  common  to  two  linear  complexes.  Hut  all  lines  which  belong 
to  the  two  complexes  defined  by  equations  (1)  belong  also  to 
all  complexes  of  the  pencil 


and  the  congruence  can  be  defined  by  ;uiv  t\\'o  complexes  obtained 
l>v  j^ivin^  X  two  values  in  ('2). 

A  complex  defined  by  (-)  is  special  when 


that  is,  when         7;  (  n  )  4-  -  X?;  (/r,  tf  )  4-  >-'-';/  (  /3)  =  0.  (:->) 

In  general  equation  (•>)  has  two  distinct   roots.    Hence  we  have 

the  theorem  : 


The  two  tixfd  lines  are  called  the  *///vr/Y/'Vx  ol  the  congruence. 
The  directrices  are  evident  conjugate  lines  with  respect  to  any 
nonspecial  complex  defined  by  equation  (-). 

If  the  roots  of  equation  (:>)  are  eijuah  the  congruence  has  onlv 
one  directrix  and  is  called  a  n^i'i-'nil  >•,,,/,/,•><< //<•<•.  This  congruence 
consists  of  lines  which  intersect  the  directrix  and  also  belong  to 
a  nonspecial  complex.  It  is  clear  that  the  directrix  must  be  a 
line  of  this  nonspecial  complex,  for  otherwise  it  would  have  a 
conjugate  line  and  the  congruence  would  be  nonspecial.  Hence 
,/  ttj>,-i-i,il  i-nni/riii'/n'i'  1'n/ixt'x/x  nl'  I'nn-x  //•///>•//  itifi'/'si-i-f  <i  fi.n'i/  liiii'  <nnl 
Ktir/i  tJnit  tjirniiijli  mi//  [>n hit  "/'  tin1  f/.ri'i/  Inn'  //"rx  <i  /><'>!,•//  /,('  ,'<>)>- 
i/rni  in-,'  ////f.v,  f//,'  t/.i'i'il  li/n1  fn-ftn/  hi  nil  riisi-s  <i  Ihn'  <>f  fin-  //*•//<•//. 

As  the  vci'tex  of  the  pencil  moves  alonu;  the  directrix,  the  plane 
of  the  pencil  turns  about  the  directrix. 

\\  e  have  seen  that  a  nonspecial  congruence  mav  be  defined  by 
its  directrices.  If  the  directrices  intersect,  the  congruence  separates 


LINK  rnoUlHNATKS  ;;o:j 

into  two  sets  of  lines,  one  beiii'_r  ;dl  lines  in  the  plane  of  the  direc- 
trices (a  congruence  of  lir>t  order  and  zero  da>s  ).  and  the  other 
bein^  all  lines  through  the  point  of  intellect  imi  of  the  directrices 
(a  congruence  of  /.ero  order  and  lirst  class  ). 

\\'hen  the  directrices  do  not  intersect,  the  congruence  is  one  of 
lirst  order  and  tir.st  cla». 

139.  The  cylindroid.  \\'e  have  seen  that  every  linear  complex  has 
an  axis.  In  a  pencil  ot  linear  complexes  •^•i\eii  bv  equation  (  '2  }. 
$]'•}*,  there  are,  therefore,  /.  '  axes  which  form  a  surface  called 
a  <•/////><//•"<</.  \Ve  mav  tind  the  equation  of  the  cylindroid  in  the 
following  manner  : 

Let  us  take  as  the  axis  <>/  the  line  which  is  perpendicular  to 
the  directrices  of  the  two  special  complexes  of  the  pencil,  as 
the  origin  n  the  point  haltwav  between  the  t\\o  directrices.  a>  the 
plane  X<>\  the  plane  parallel  to  the  two  directrices,  and  as  <>.\ 
and  <>Y  the  lines  in  this  plane  which  bisect  the  angles  between 
the  two  directrices.  That  is,  we  have  so  chosen  the  axes  ot  refer- 
ences that  the  equations  of  the  two  directrices  of  the  special 
complexes  of  the  pencil  are 

I/  —  nt.r  =  0,  .:  =  '•,  (  1  i 

and  //  +  iiu'  =  (l<  —  '%  ( -  ) 

respectively. 

The  I'lueker  coordinates  of  the  line  (1  ).  which  may  be  deter- 
mined by  the  points  (  u,  I),  ,•)  and  (  1.  m,  /•),  arc 

and  the  special  complex   \\ith  this  axis  K  therefore,  bv  (  1  ).  ^  1 1".'. 

Similarly,  the  coi'irdinates  of  (  •_! )  are 

/'',•.'    -  0,      )>''2}=/;      //,-j        -1,       /-.,-        -///••.      /',•'       -  in.      l'\=(\ 
and  the  special  complex   with   this  axis   is 

The    pencil    of   complexes    is    therefore 


324  FCH'R-DIMKNSIOXAL  GEOMETRY 

My  (•'))'  £  l:*n''  tin1  equations  of  the  axis  of  any  complex  of  the 
pencil  are 

(1       X  )  ///  :  —  (  1  4-  X  )  tits  _  -(]  +  X  )  z_+(  1  —  M  •  • 
-   (  1  f  X)  -(1-  X)>/1 

-  (  1  -  x  )  "<•>'  -O  +  X)  y 

u 
l-x 

\vllich   reduce  to  i/  —    -  >iu; 

1  -f-  X 

[(  1  -  X  )•//<•  +  (  1  4-  X)'-']  *  =  (  1  -  X-)  (  1  4-  in'1)  c. 
It'  we  eliminate  X  from  these  equations,  we  have 


which  is  the  required  equation  of  the  cylindroid. 

The  equations  slmw  that  the  surface  is  a  cnhic  surface  with  <)/ 
as  a  double  line.  All  lines  on  the  surface  are  perpendicular  to  OZ, 
and  in  any  plane  perpendicular  to  <>Z  there  are  two  lines  on  the 
surface  which  are  distinct,  coincident,  or  imaginary  according  as 
the  distance  of  the  plane  from  (>  is  less  than,  equal  to,  or  greater 

(1  -f  in')'- 
than 

_  /// 

\Ve  may  put  the  equation  of  the  cylindroid  in  another  form.  We 
shall  denote  by  -  <i  the  angle  between  the  directrices  of  the  special 
complexes  of  the  pencil,  by  0  the  angle  which  any  straight  line 
on  the  cylindroid  makes  with  <  L\\  and  by  /•  the  distance  of  that 

line  from  <>.    Then  ///  —  tana,  and          m  =  tan  0. 
„          .  1  +  X 

Lquation  (:>)  then  becomes 

- 


sn  - 
sin  -  a 


140.  The  linear  line   series.     Consider  three   independent  linear 

f(l  UiltlOIlS  x^  V^     >  V^ 

2/V',=  (J'  tf,-r,=  {}<          £,7^=®-  (1) 

'I  liese  equations  are  satisfied  by  the  coordinates  of  lines  which 
are  common  to  the  three  complexes  defined  by  the  individual 
equations  in  (1  )  and  define  a  line  .sv/'/Vx.  Any  line  of  the  series 
also  belongs  to  each  complex  of  the  set  given  by  the  equation 

V(  A.I  4-  ^.f  j.7  ,.,-      i).  (  -2) 


LINK  COORDINATES  :;-J.", 

and  any  three  linearly  independent  equations  formed  from  (  '_'  >  In 
giving  to  A,  /z.  and  v  definite  yalues  determine  the  same  line  series 
that  is  determined  liy  (  '2  ). 

A  complex   of  the  type  ( '_' )  is  special  when 

//  (  \n  +  /ZP1  +  ^7  )  =  A'//  (H)+  /z-';/  (  p1  )  -f  l'~i]  (  7  )  -f-  -  A/z;y  (  a.  p'  ) 

-f  !_>  /zr//  (  p\  7)  +  -2  i'\>j  (  7,  a  )  =  0.    (  :'.  ) 

There  are  a  singly  infinite  numlier  of  solutions  ot  equation  <  :'>  > 
in  the  ratios  A  :  p. :  v.  Hence  the  lines  -which  are  defined  Ky  equa- 
tioiis  (1)  intersect  an  infinite  numlter  of  straight  lines,  the  axes 
of  the  special  complexes  defined  l»y  ( '_' )  and  (  '•>  ).  These  lines  are 
called  the  i///vr//vV,  x. 

The  arrangement  of  the  directrices  depends  upon  the  nature  of 
equation  (•'>)•  In  studying  that  equation  we  may  temporarily  in- 
terpret A :  /z :  i'  as  homogeneous  point  coordinates  of  a  point  in  a 
plane  and  classify  equation  (  :» )  as  in  vj  :>.">. 

Let  us  place 

!'/(")          '/  ( ''•.  p)      '/ ( 

/>  =    >)  (n.  fi)      i]  (  tf )  jj( 

j  '/  (  a-  7  )       '/  ( /^i  7  )       '/ 1 

CASK  I.  I>  :  <J.  This  is  the  general  case.  Kqiiatioii  ('•}),  inter- 
preted as  an  equation  in  point  coordinates  A  :  /z  :  v,  is  that  of  a  conic 
without  singular  points.  To  any  point  on  this  conic  corresponds  a 
special  complex  of  the  type  ( "_' )  whose  axis  is  a  directrix  of  the 
series  (1).  To  simplify  our  equations  \ve  shall  assume  that  the 
coordinates  .<\  are  Klein  coordinates.  Then  (l»y  theorem  II.  ^  1  :'>•>) 
if  (  \i :  ti]  :  i>{ )  and  ( A., : /z., :  j'?)  are  two  solutions  of  equation  (  '2  ), 
the  axes  of  the  corresponding  special  complexes,  or,  in  other  words, 
the  corresponding  directrices  of  the  series  (  1  ),  are  A.,'t.+  /z1p)i  f  i'}y 
and  \._,n  -f-  /z.^(  +  ''-7, . 

The  condition   that    tliese  t  \\  o  direetrices   intersect    is 


which  is  exact  1\  the  same  as  the  eomlit  ion  that  each  of  t  he  two  points 
(  At :  /i  :  i>  )  ami  (  A., ://,:;'.)  si  i  oil  li  I  lie  on  t  he  polar  of  t  he  other  \s  it  h 
I'espect  to  the  conic  (  '•}  ).  Thi.^  is  illlpossil lie,  since  each  ol  the  points 
lies  on  t  he  conic.  It  t'ollou's  from  this  t  hat  //•<  f/i'n  ,//',••  <•//•/'•>  x  <//?>  /'M  <-t. 
I-'roin  this  it  will  al>o  follou  that  im  t//'»  ////IN  ••/'///«  ;///•-//  xirim 
t/ttffxi-i-t,  tor  it  the\  did  each  directrix  mu>t  cither  lie  in  their 


plane  or  pass  through  tlu'ir  common  point,  and  some  of  the 
directrices  \vould  intersect. 

The  lines  of  the  series  (1).  on  the  one  hand,  and  their  direc- 
trices. on  the  other,  torm.  therefore,  t\vo  tanniies  ot  lines  such  that 
no  two  lines  ot  the  same  tamilv  intersect.  Inn  each  line  ot  one 
family  intersects  ail  line.-*  of  the  other.  This  surest  s  the  two  fam- 
ilie-*  of  generators  on  a  (jiiadric  surface.  That  the  configuration 
is  ivalK  that  of  a  (jnadrie  surface  follows  trom  the  theorem  that 
the  locus  of  lines  which  intersect  three  nonintersecting  straight 
lines  is  a  ijnadric  surface  (see  Kx.  i>,  p.  -\'2~  ). 

We  sum  up   in   the  following  words: 

III      till'     </'ll'   Till     fllKf     (    />     ---    0)     //,,'      ///H'K      //'///'  7/      tlt't'      I'll///  Hint/      fn      tJu'll' 

liiti'ir  f'niiiiift'j'cx  I'l'i'm  "ii>'  I'liinilii  at'  i/i  m  t'ltfnrx  at'  <t  ijHiii/t'if  xii  rfii'-f. 
tin  i  r  ill  r,  ••(  i-t'-,K  J  'iriiiini/  tin'  xii'iiiiilj'tiittli/. 

A    family  of  generators  of   a  (jiiadric  surface  is  called  a  /Y///////N. 

CASK  II.  1>  —  0.  hut  not  all  the  lirst  minors  are  /ero.  The  curve 
of  second  order  detined  l>v  ('•})  reduces  to  two  iiitt'i'secting  straight 
lines  and,  liv  a  linear  substitution,  can  lie  reilueed  to  the  form 

Xj.  =  0. 


These  are  t  hree  special  com- 
plexes such  that  the  axes  of 
the  tirst  two  dd  not  intersect. 
hut  the  axis  of  the  third  inter- 
sects  each  of  the  axes  of  the 
tirst  two.  The  axes  lie.  there- 
fore, as  in  Fi'4\  -Vs.  The  series 

consists,  therefore,  of  two   peli- 

...  .         .  I  i  '  •  .  ">s 

ells  ot    lilies  :    one    Ivilig  in   the 

plane  ot  -/  and  <•.  with  its  vertex  at  /•''.  the  point  of  intersection 
ot  I,  and  •:  the  other  1  \in-_;'  in  the  plane  of  /•  and  '•.  \\ith  its  vertex 
at  /•'.  the  intersection  of  </  and  <-. 

CAST:  III.  l>  n.  all  tin-  lir>t  minors  are  /.ero.  hut  not  all  the 
second  minors  arc  /ero.  The  conic  deiincd  !i\  (  :J  )  consists  of  t  \\  o 
coincident  lines.  It-  equation  ma\'  he  made  r"  ". 


LINK  COORDINATES  ;;-J7 

\\e  have  thru  taken  to  delhie  the  series  three  complexes  of 
which  t\vo  arc  special  with  intersecting  axes,  and  the  third  is  non- 
special  and  contains  the  axis  of  the  other  two. 

If  <i  and  />  are  the  two  axes  of  the  special  complexes,  /•'  their 
point  of  intersection,  and  ///  their  common  plane,  then,  since  the 
nonspeciul  complex  contains  <i  and  ?>,  /•'  is  the  pole  of  m  with 

respect     to     that     complex.      Ilelice     the     lilies     common     to     the     two 

complexes   torm  a  pencil   ot    lines  which   must    be   taken   double   to 
preserve  the  order  ot   the  complex. 

(  'ASK  I  V.  The  case  in  which  all  the  second  minors  of  I >  \  am-h  is 
inadmissible,  for  in  that  case  the  three  complexes  in  (  1  )  arc  special 
and  their  axes  intersect.  Then,  from  £lol,  7,=  <l,+  'V^<  imd  the 
three  equations  (  1  )  are  not  independent. 

EXERCISES 

Two  complexes  A,",./',  =  0  and  ]5yV,  —  "  ;llv  Hl  Ini-nlutlnn  when 
rj(n.  !•}=  0. 

1.  Prove  that   it  //  is  a  line  common  to  two  complexes  in  involution 
the  correspondence  of  planes  through  //,  which  can  he  set   up  \<\  taking 
as  corresponding  planes  the  two  polar  planes  of  cadi   point   of  /.  uith 
res]ieet  to  the  two  complexes,  is  an  involution. 

2.  Prove   that    two   special    complexes   are    in   involution   when  their 

axes    intersect. 

3.  Prove  that   a   special   complex    is   in    involution   with    a    noiispecial 
complex   when  tin-  axis  of  the  former  is  a  line  of  the  latter. 

4.  Prove   that    if  two   iioiispeeial   complexes  are    in    involution  there 
exist   two  lines,  i/  and  //.  which  are  conjugate  with  respect   to  the  two 
and  such  that   the   polar  planes  of  an\    point   /'  are  harmonic  conjugates 
with    respect    to  the   t  \\  o   planes   through    /'  and  <i  and    throiiuli   /'  and   It 
respectively,  and  also  such   that   the  poles  of  anv   plane  ///   uith  I'espect 
I  o  the  i  \\  o  complexes  are  harmonic  con  ju^ates  to  t  he  j  mi  nt  s  in  \\  h;ch  /// 

meets   ,/  and    li . 

;>.  Prove  that  the  six  complexes  ./•  0.  where  ./'  are  Klein  coordi- 
nates, are  two  li\  two  in  involution.  Ilencc  prove  li\  a  I  raiistormat  mn 
"t  coordinates  that  there  exists  an  intinile  numhei-  of  siu-li  sets  ,.(  six 
coin  pic xe>  i n ut  ual  1  v  in  ins  ol ut  ion. 

•i.      Prove    that     the    locllS    of    Hlles    which    illtel'SCct    three    I  id  1  1 !  1 !  e  r-eet  1 II  - 

1  nies  i>  a  i  jiiai  1  ric  surface,  \*\    11^1  ii'j,   Plueker  coord  i  nal  es  and  I'liin  mat  in-_r 

one    set    ot     point    (Mil  il'il  mates. 


:}-2S  rori;  DI.MKNSIONAL  GEOMETRY 

141.  The  quadratic  line  complex.    A   quadratic    line   complex   is 
defined  l>v  an  equation  of  the  form 


We    shall    consider  only   the   genera!    case    in    which  the  above 
equation   can   be   reduced  to  the  form 

2'V*7=0,          ('-,-")  (1) 

at    the   same    time    that    the    coordinates  J\  are   Klein   coordinates 

satisfyin     the  fundamental  relation 


Let  us  consider  any  lixeil  line  yt  of  the  complex  and  any  linear 
complex  ^-\ 

2/Vi--0'  (•*) 

containing  if..  In  general  the  complex  (8)  will  have  two  lines 
through  any  point  /'  in  common  with  (1),  for  /'  is  at  the  same 
time  the  vertex  of  a  pencil  ot  lines  of  (  -\  )  ami  of  a  cone  of  lines 
of  (1  ). 

Analytically,  we  take  /',  a  point  on  //,,  and   r_,  anv  line  of  (  •>  ), 
but  not  of  (  1),  through  /'.     Then  anv  line  of  the  pencil  determined 

by  (f   and  r    is 

pj\  =  //,  +  \zt, 

and  this  line  always  belongs  to  (  o  ),  but  belongs  to  (1)  when  and 
only  when  _ 


This  i;ivcs  in  general  two  values  of  X,  of  which  one,  A.  =  0,  deter- 
mines the  line  //f  and  the  other  determines  a  different  line.  lint 
the  two  values  of  \  both  become  /ero,  and  the  line  //i  is  the  onlv 
line  through  /'  common  to  (  1  )  and  (  -\  )  wlien 

2'>,.^,=  0; 

that  is,  when   ^t  has  been  chosen  as  anv  line  of  the  linear  complex 


//;    t/iift    '•'/>•'•    thf   /'"/'//•  /'f'l/if    "f  I'    H'ttlt    fi'f/n-rf    fu   (  4  )    /.<    tdii'/fht    t" 
th>     '••>iii  i>l>  J'    '-"ii>'    "f   (  1   )    <lt    /',    IvJuTi'    I'    I*    ilitij    i>"tnf    t/'fttltt'l'i'f   lit'    l/t. 

The   complex   (  I  )   is   accordingly  called   the  /<»/;/'  //f  l/nnir  m////!/,-./- 
/'   n  .     It    i<  often  said   that   the  tangent    linear  complex  contains  all 


LINE  COORDINATES 


lines  of  tlic  complex  (  1  )  which  ;irc  consecutive  to  //,,  since  any  line 
with  coordinates  //,+  <///,  sat  islies  (  4  ).  The  discussion  we  have  given 
makes  this  notion  more  precise. 

More   generally   we   have  at   //,  a   pencil    of   tangent    linear  com- 
plexes.    For  by  virtue  of  (-)  the  complex   (1)  may  lie  written 


where  fj.  is  any  constant,  and  the  tangent   linear  complex   to  (">)  is 
2  K  +  AO  .'/,-'•.-    "•  («'') 

All  these  complexes  have  the  same  }  tola  r  plane  at  any  point  /'of  //,. 

If//,  is  not  a  line  of  the  complex,  equation  ('•)  defines  a  pencil 
of  imhir  Itni'iir  <•<'  ni  nit'.  ft'*. 

'I'lie  line  //,  is  called  a  xini^ilai'  li/u1  wlieii  the  tangent  linear 
coiiijilex  (4)  is  special.  The  condition  for  this  is 

2/7,'/?=0,  (7) 

which  says  that  <•,//,  are  the  coordinates  of  a  line,  the  axis  of  the 
tangent  complex.  At  the  same  time  all  the  complexes  (  ti  )  are  special 
and  have  t  he  same  ax  is. 

This  axis  intersects  //  .  since  ^\  '///?  =  "  (because//,,  is  a  line  of  the 
complex),  and  the  intersection  ot  the  two  lines  is  called  a  ttitu/ulttr 
/<"////,  and  their  plane  a  tit'ttt/itlnr  jilmn'.  Anv  complex  line  v,  f°'' 
which  condititin  (7)  holds  is  called  a  ximjnliir  line. 

Let  /'  be  a  singular  point  on  a  singular  line  //,,  let  z.  be  anv  line 
through  /',  and  consider  the  pencil  of  lines 


The  condition   that   ./'_  belong  to  (1  )  is 


since  ^>  cjj'f  --  1  1,  because  //.  is  on  (  1  ),  and  N  ciyi.^i=0,  because  zi 
intersects  (•__//.  at  /'.  Then  if  zt  is  a  line  of  (  1  ).  all  lines  of  the  pencil 
(  s  )  belong  to  (1  ).  (  )n  t  he  ot  her  hand,  if  zi  is  anv  line  not  hulon^injjf 
to  the  complex  (  1  '),  tht'  line//,  is  the  oiilv  line  in  the  plane  (///,) 
\\hieh  belongs  to  the  complex.  This  makes  it  evident  that  <tt  << 

MHi/Hftt/'  />"//lt  t/li'  fninf'li'.f  I'nili'  Xjilitx  Ufi  illtn  f  //'"  fjitllt'  fH'tlfiltl 
tllti  TKi'i'ti  ii'J  III  tin'  MHi/ulllt'  //Hi'. 


Koril   DIMENSIONAL  GEOMETRY 


In  a  similar  manner  we  may  take  j>  as  a  singular  plane  through 
a  singular  liiu1  //,,  zt,  anv  line  in  />  inU'i'seetiug  //,,  and  again  con- 
sider the  pencil  (v's).  \Vc  ohtain  again  (il),  hut  the  interpretation  is 
now  that  it  .:,  is  anv  complex  line  in  /»,  there  is  a  pencil  of  lines  in 
i>  \vith  vertex  on  i/t.  Consequently  in  t<  Miiifuhti'  [iliine  the  complex 
emi'ie  Ki'lifK  ii  [>  ///t"  tn'n  ftfticila  f<>  which  tin'  Kiiii/nttif  line  tx  eninin<»i. 

\\'e  shall  now  show  that  (tiuj  /»>///!  *</  trfiic/i  the  cuniplt'jc  cmie 
x[ilitn  int"  tii'n  pencil*  ix  il  x//i</u/t<r  [mint  ttml  iini/  i>/ti/n-  in  which 
(/<>•  I'niinih'.i'  i'ii/uf  xplita  tut  (>  (/('i>  jit'tictlu  /*  <t  Ktnifuldr  pliuit1, 

Let  ./  lie  such  a  point,  and  let  the  two  pencils  he  </_  -f-  \!>t  ami 
(/  +  ''  .  'I'hell 


The  tangent  complex  at  </(  contains  iif,  lt.  and  c  :  hy  (1").  Tliere- 
t'nre.  liv  theorem  V,  ^  1  '-\'-\,  it  is  special,  and  the  point  A  lies  on  its 
axis.  Hence  A  is  a  singular  point.  The  second  part  of  the  theorem 
is  similarly  proved. 

Now  let   '',  and  /'_  he  two  intersecting  complex   lines.     Then 

V,  ,',:(),          V/;-=0,          y<f/,.=    U,  V,  .„-::(),  V  ,./,-=().(    1   ]     ) 

-        ,  ^      t  S-4       '      '  —  '      '       '  ^      '      ' 

It'  the  pencil  tit  -f  X//(  hi'longs  entirely  to  the  complex  we  have  also 


We    shall    fix    <i:   and    take    as    lt   that    line    of    the    pencil    whit 
intersects  a  tixed  line  </   which  does  not   intersect  <i  . 


To  determine  A(  we  have  live  equations  of  which 
three  are  linear  and  two  quadratic.     There  are  there- 
!<>re   in    general    lour  sets  ot    values  of  // ,  so  that   <-/?      .1 
<in  if    line    /if     t  lie     mill  file.!'     tin  re    lire    in    ijem  T'il  f"iir      | 
xnn/nliir   i>',tiitx. 

Let  the  four  points  he  .1  .1 ,.  .(..,.•!(  I'V.  .">'.»  )  and 
the  four  lines  lie  A',  I",  I"1.  I"".  Then  each  of  the 
planes  (  iif>'  ).  (  -/A  '  ),  (  ,//,'"  ).  (  nl,""  )  coiitain.s  a  peiieil 
ot  lines  mid  lu-1  ice  a  second  one  (list  inet  or  coincident. 

Tlh'l'efoiv  ////•"  n  i  ili  1 1  ii  il  line  <>n  fin    cumn/cj'  t  In  re  tl/'i    t'uiir  xi  in/nlii  r  fililin  x. 

Since    the    coi'irdinates    of    the    four    lines   /     sat  i>f\'    three    linear 
equations,  the   lines  li.-loiig  in   general  to  \\.   rr^iilus  (  ^  1  1")  and  do 


LINK  COORDINATES  331 

not  intersect.  Therefore  the  tour  points  .1  arc  in  general  distinct, 
as  are  the  four  planes  ('//-)•  1"  order  that  two  points  or  planes 
should  coincide  it  is  necessary  that  the  re^ulus  should  degenerate, 
as  in  ( 'use  II,  ^  1  4<>.  The  condition  tor  t  his  is  that  the  discriminant 
of  the  equation 

x-v,r+M-y,/;+-z;-v,-v-f  L'x/zVw+i^v^/^+^xy,. _,,*=<) 

should  vanish.  P>v  virtue  ot  (11),  and  the  fact  that  </t  satisfies  ( '2  ), 
the  ahove  equation  reduces  to 

,,-2\or+  -JX/iV,,,/  +-  iV']^, '•/','/,=  0; 
and  the  condition  that    its  discriminant  should  vanish  is 

since  V,/ r,7  =£  (I.  liv  (  1 :> ). 

If  this  condition  is  met,  ",  is  a  singular  line  hv  the  previous 
definition,  two  of  the  points  A  ,  ./„,  ./.,.  ./  coincide  into  one  sin- 
gular point  on  '/(,  and  two  ot  the  singular  planes  coincide.  More 
prccisclv,  if  ./  and  ./.,  coincide  at  .1  the  pencils  (<if'')  and  («!>") 
form  the  complex  cone  at  .1.  the  two  lines  //"  and  !>""  intersect  on  </ 
(compare  ^  1  1(>).  and  the  points  .1,  ami  .1  are  the  vertices  ot  the 
pencils  of  complex  lines  in  the  plane  ('//<"'//""). 

142.  Singular  surface  of  the  quadratic  complex.  The  singular 
points  and  planes  are  determined  l>v  the  complex  line  >/•  and  the 
intersecting  line  '•_//,  \\hcrcN  ,-'-'_//-  —  <). 

\\'e   take   the   pencil 

Then  .v_  satisfies  the  equations 

V  =V//;=    (».         v      ''      .,  z;  =  V  >'„>/;  =  0; 

or,  what    amounts  to  the  same  tiling,  the  equations 

V      '       ,f  =  0  V         '          ,-'       n.  ,|, 

— -•,  H\  '  —!••,  +  ^r 

K([iiation  (  1  )  shows  that    .?    is  a  singular  line  ot   the  complex 

V      1       ,.-^0.  i'2) 


FOTi;    DIMENSIONAL   (JKOMKTKY 

Since    tin1    lines   z{   and  belong   to   thr   same   pencil    as  //; 

'',  ~(~  X 

and  '•,'/,.  ''"'  singular  points  and  planes  of  ('!)  are  the  same  as 
those  of  ^'V,:  =  "•  Il(l  matter  what  the  value  of  X.  The  com- 
plexes ( '_' )  are  called  rnMni/uliir  '•"////'/'•.rex. 

\\'e  mav  use  the  cosingular  complexes  to  prove  that  <>»  </////  Inn' 
in  sp<t<'t'  In'  four  xnn/iiliir  points  of'  tin1  I'oinj'Ii'.t'  ^  «y~  =  0,  a  ml  thi'oinjh 
tin  if  lint'  i/o  four  siinjuliir  pining. 

Let  /  he  any  line  in  space1.  We  may  determine  X  in  (-2)  so 
that  /  lies  in  the  complex  (-):  in  tact,  this  mav  l>e  dune  in  four 
wavs,  since  (-)  is  of  the  fourth  order  in  X  l>y  virtue  of  the  relation 

V\/--  =  0.  Then  there  will  he  four  singular  points  of  this  new  com- 
£~i  ' 

plex  on  /  hy  previous  proof,  and  these  points  are  the  same  as  the 
singular  points  of  ^  -yr  =  0. 

It  follows  at  once  that  the  lorns  of  the  sint/uliir  points  of  it  //n<i<]- 
rnfi''  i'wnitlej'2  <-:.r~  =  0  /x  <i  si(.rf<i<'f  <>f  tin'  fourth  nri/i-r.  <nnl  the 
enrt'lopt'  of'  tin1  x'nnjulnr  plane*  is  n  snrfiii-f  of  ///,•  fmo'th  fliis*. 

Tln'Sf  tit'o  surfiict's,  Jtoii'ci't'i'.  ni'i'  fht1  sti/iH'  sn rfiii'i'.  I''or  if  two  of 
the  singular  points  on  /  coincide,  two  of  the  singular  planes  through 
I  also  coincide.  Therefore,  if  /  is  tangent  to  one  of  the  surfaces  it 
is  tangent  to  the  other.  Hut  /  is  any  line.  Thei'efore  the  two  sur- 
faces have  the  same  tangent  lines  and  therefore  coincide. 

This  surface,  the  locus  ot  the  singular  points  and  the  envelope 
of  the  singular  planes,  is  called  the  nin;iuJnr  snrf <(<•>'. 

We  shall  not  pursue  further  the  study  of  the  singular  surface. 
Its  ('artesian  equation  may  he  written  down  !>v  first  transform- 
ing from  Klein  to  IMitcker  coordinates  and  replacing  the  latter 
hv  their  values  in  the  coordinates  of  two  points  (./•,  _//,  ,r)  and 
(./•',  //'.  .:' ).  Then,  if  (./•',  //',  z' )  is  constant,  the  equation  is  that 
of  the  complex  cone  through  (./'',  //'.  ,:').  The  condition  thai  this 
cone  should  degenerate  into  a  pair  of  planes  is  the  ('artesian  equa- 
tion of  the  singular  surface.  It  mav  he  shown  that  the  Mil-face 
has  sixteen  doiihle  points  and  sixteen  douhle  tangent  planes 
and  is  therefore  identical  with  the  interesting  surface  known  as 
k  uimuer  s  surface. ' 


LINK  COOK  DIN  AXES  3:-J:-J 

EXERCISES 

1.  Prove  that    the   tangent   lines  of  a   fixed   quadric  surface  form  a 
quadratic  complex.     Find  the  singular  surface.     Note  the   peculiarities 
when  the  quadric  is  a  sphere. 

2.  Prove  that  the  lines  which  intersect  the  four   faces  of  a  fixed  tet- 
rahedron in  points  whose  cross  ratio  is  constant  form  a  quadratic  com- 
plex   whose    equation    may    be    written    .  l/',,/'^  +  /;/'i.t/'4-  +  ' '/'u/'^  —  °- 
This    is   the    trt,',ij,r<lrnl  rin,i/,lf.r. 

3.  Prove  that   in  a  tetrahedral  complex  all  lines  through  any  vertex 
or  Iving  in  anv  plane  of  the  fixed  tetrahedron  belong  to  the  complex. 
Find  the  singular  surface. 

4.  Show  that  lines,  each  of  which  meets  a  pair  of  corresponding  lines 
of  two  protective  pencils,  form  a  tetrahedral  complex. 

5.  Show  that  the  lines  connecting  corresponding  points  of  a  collinea- 
tion  form  a  tetrahedral  complex. 

6.  If  the  coordinates  of  two  lines  ,r,  and   //,   are   connected  by  the 
relat  ions 

p.i\  = 


V,,  +  x 

show  that  .>•,•  belongs  to  the  complex  5/',3*,"  =  0  and  that  //,  belongs  to 
t  he  cosingular  complex 


1.  If  j\  and  ./•'  are  two  lines  of  a  complex  (',  and  //-  and  //'  their 
corresponding  lines,  as  in  Hx.  o,  of  a  cosingular  complex  ( \,  prove  tin1 
following  proposit  ions  : 

(  1  i    If  ./•,  intersects  //'.  then  ./•'  intersects  //,. 

( !_' )  If  .r,  intersects  ./•'  at  /',  and  //,-  intersects  //'  at  (j,  the  complex 
cone  of  ('  at  /'  and  the  complex  cone  of  C'A  at  (>  degenerate  into  plane 
pencils,  and  to  a  pencil  of  either  complex  corresponds  a  pencil  of 
the  other. 

i  •>  i  It  ,/•,  intersect s  ,/•'  at  /',  ill  general  y,  does  not  intersect  >/'.  and  the 
complex  cone  of  ('  at  /'  i-orrcsponds  to  a  regulus  of  <  \.  Also  the  com- 
plex conic  in  the  plane  of  ./•,  and  ./•'  corresponds  to  a  re^uliis  of  <  'A. 

i  1  i  Anv  t  wo  lines  ,rt  and  ./•'  of  ( '  which  do  not  int  ersect  determine  a 
cosingular  complex  f\  in  which  the  two  lines  ,/i  and  //',  corresponding 
to  r,  and  ./•'.  intersect .  There  are,  t  he  re  fore,  two  ivguli  of  <'  t  hn  aigh  ./•, 
and  ,r\  corresponding  to  the  complex  cone  and  the  complex  conic  of  (\ 
determined  1>\'  i/:  and  //'. 


:}:}\  For  u-  DIM  KNSK  >N  A  L  <;  K<  >M  KTII  v 

8.  Prove  that  for  an  algebraic  complex  ft  .r},  .r,,  r.(,  .r^,  ./•.,  .'',.)—  0  of 
the  decree  n  the  singular  lines  are  given  liy  the  equations 

/,.=«,  S(;;;;=o, 

and  that  the  singular  surface  is  of  degree   ~»(n  —  1)-,  where  singular 
line  and  surface  are  defined  as  for  the  quadratic  complex. 

143.  Pliicker's  complex  surfaces.  In  any  arbitrarily  assumed 
plane  the  lines  which  belong  to  a  given  quadratic  complex  envelop 
a  conic.  If  the  plane  revolves  about  a  fixed  line,  the  conic  describes 
a  surface  called  by  IMiicker  a  nu'ritlliin  xnrt'<t<r  of  the  complex. 
If  the  plane  moves  parallel  to  itself,  the  conic  describes  a  sur- 
face called  bv  IMiicker  an  fi^nnt'trinl  xnrf<ifi'  of  the  complex.  It  is 
obvious  that  an  equatorial  surface  is  onlv  a  particular  case  of 
the  meridian  surface  arising  when  the  line  about  which  the  plane 
revolves  is  at  infinity.  In  either  case  the  surface  has  been  called  a 

It  is  not  difficult  to  write  down  the  equation  of  a  complex  sur- 
face. Let  the  line  about  which  the  plane  revolves  be  determined 
bv  two  fixed  points,  .1  and  />',  let  /'  be  anv  point  in  space,  and  let 
ut  and  r.  be  the  coordinates  of  the  lines  /'./  and  /.'/'  respectively. 

Then  the  coordinates  of  anv  line  of  the  pencil  defined  bv  /'.I 
and  /'/.'  are  //,  -+-  X'*,,  and  this  line  will  belong  to  the  quadratic 
complex  "V <•,./•,"  =  (|  when  X  satisfies  the  equation 

In  general  there  are  two  roots  of  this  equation,  corresponding  to 
the  geometric  fact  that  in  anv  plane  through  a  fixed  point  there 
arc  onlv  two  complex  lines,  the  two  tangents  to  the  complex  conic 
in  that  plane.  If.  however.  /'  is  on  that  conic,  the  roots  of  (1) 
must  be  equal  :  t  hat  is 

Now  ?/,  involves  the  point  coordinates  of  .1  and  /'  lincarlv.  and 
>•.  involves  in  a  similar  manner  the  coordinates  of  /.'  and  /'.  Hence 
(  '1  )  is  of  the  fourth  order  in  the  point  coordinates  of  /'. 

From  the  construction  /'  is  anv  point  on  the  complex  surface 
formed  bv  the  revolving  plane  about  the  line  .//•'.  Hence  Pli'n'ki'r'n 
,•', in nl ,  .r  xn /'tiiri'ft  ,/,-,'  nt  ///c  fuiirtli  H/-I/,  r. 


LINK  COORDINATES  :;;;:, 

We  mav  work  in  the  same  way  with  plane  coordinates;  ilia! 
is,  we  niav  define  a  straight  line  l>y  the  intersection  of  t  \\  u  lixed 
planes,  't  and  /^,  and  take  .!/  as  anv  plane  in  spare.  '1  hen  the  three 
planes  fix  a  point  on  /,  and  equation  (  ]  )  del  ermines  the  two  hues 
through  that  point  in  the  plane  .!/  which  belong  to  the  ipiadratie 
complex.  Hence,  it  t  he  em  »rd  mat  es  ot  M  sat  1st  v  e<  |  nat  ii  >n  ('!).  M  is 
tangent  {n  the.  complex  cone  through  that  point  mi  /.  A  little 
reflection  sho\\'S  that  such  a  plane  is  tangent  In  the  ruinplrx  siir- 
taee  formed  l»v  revolving  a  plane  about  the  line  /  and  that  anv 
tangent  plane  tn  the  complex  snrtaee  is  tangent  to  a  cone  o|  com- 
plex  lines  \vith  its  veil  ex  on  /.  Hence  (  '2  )  is  t  he  cipiat  ion  in  plane 
coordinates  ot  the  complex  snrtaee.  I'herelore  <t  <•<,,//  ji/r.r  xiirt'ufi' 
!x  <>f  tin'  fourth  'V'/x.v. 

144.  The  (2,  2)  congruence.    (  'onsider  the  eon^nieiiee  detined  1>\ 

the  t\vo  eiiiiations 

V,,,  ..._-o,  (1, 


\vlneh    consists    ot     lines    common    to    a    linear    and    a    Quadratic 

i 

complex.  Through  everv  point  ot  space  <4~o  t\vo  lines  of  the  con- 
^nience  :  namely,  those  common  to  the  pencil  of  lines  of  (1  )  and 
the  complex  cone  ot  (  ;>  )  through  that  point.  Similarly,  in  even 
plane  lie  two  congruence  lines  whirl)  arc  common  to  the  pencil 
of  (  1  )  and  the  ennie  of  ('2)  in  that  plane.  The  complex  is  there- 
lore  ot  second  order  and  second  class  and  is  called  the  (  '_'.  '2  ) 
congruence. 

('onsider  anv  line  _//.  ot  the  congruence,  and  /'  anv  point  on  it. 
Through  /'there  will  i;'o  in  an  exceptional  manner  oiilv  one  con- 
gruence line,  when  t  he  polar  plane  ot  /'with  respect  to  (  1  )  coincide-* 
with  the  polar  plane  ot  /'  with  respect  to  the  tangent  linear  ctnu- 
plcx  ot  (  2  )  at  //,.  This  will  occur  at  two  points  mi  //  .  This  ma\ 
he  seen  without  Jllialvsis  trom  the  tad  that  to  e\  er\  point  on  // 
ma\  he  assoeiat  eil  t  \\  i  >  planes  through  >/,  :  nanicK  .  t  he  |  mlar  ]  ilaiies 
with  respect  to  (  1  )  and  to  the  tangent  linear  complex  at  '/  .  I  Iciice 
these  planes  are  in  a  one-to-one  correspondence,  and  there  arc  two 
fixed  points  ot  such  a  correspondence. 

Analytically,  if  the  complex  (  1  )  and  the  tan^eiil  linear  complex 
ot  (  '2  )  have  at  /'anv  line  :  in  common  di>tinct  from  <i  .  thcv  \\ill 


Fot'R   DIMENSIONAL  GEOMETRY 

have    tlie    entire   pencil   //.  4-  X<~,-    in    common.     The    conditions   for 
this  are 


This  determines  a  line  series  which,  by  ^140,  degenerates  into 
two  plane  pencils  with  vertices  on  //,.. 

The  points  on  //,  with  the  properties  just  described  are  called 
the  food  i><>infa  /^  and  /•',  ol  //(,  and  the  planes  of  the  common 
pencil  of  (1)  and  the  tangent  linear  complex  of  (-5)  are  called 
the  food  />/<i)t<'x  f^  and  _/',.  The  focal  points  are  often  described 
as  the  points  in  which  //,  is  intersected  by  a  consecutive  line.  The 
meaning  of  this  is  evident  from  onr  discussion.  For  at  /•'  and  /•'„ 
the  pencil  of  lines  of  (  1  )  is  tangent  to  the  complex  cone  of  ('2),  so 
that  through  /•'  or  /•',  goes  onlv  one  line  of  the  congruence  donblv 
reckoned. 

The  locns  of  the  focal  points  is  the,  food  xnrf<io:  It  will  bo 
shown  in  the  next  section  that  the  line  //(  is  tangent  to  the  focal 
surface  at  each  of  the  points  /^  and  /•',,  and  that  the  planes  /j  and 
_/',  are  tangent  to  the  same  surface  at  /•',  and  /-^  respectivelv. 

145.  Line  congruences  in  general.  A  congruence  of  lines  consists 
of  lines  whose  coordinates  are  functions  of  two  independent  vari- 
ables. For  convenience  we  will  return  to  the  coordinates  tirst 
mentioned  in  sj  Il27  and,  writing  the  equation  of  a  line  in  the  form 


will  take  r,  .*>•,  p,  and  rr  as  the  coiirdinates  of  the  line.  Then,  if 
/•,  .?,  p.  a  are  functions  of  two  independent  variables  n.  (3,  the  lines 
(  1  )  form  a  congruence. 

Let  /  be  a  line  of  the  congruence  for  which  n       n^,  $    -  /^  .     If  we 

place 

ft  =0(«),  (2) 

we  arrange  the  lines  into  ruled  surfaces;  and  if  we  further  impost? 
on  (f)  (  n  )  the  single  coiidit  ion 


LINK   r<  ><">!;  1HNATKS  337 

we  shall   have  all   ruled   surfaces  which  are   formed  of   lines  of  the 
congruence  and  which  pass  through  /. 

It  is  desired  to  know  how  many  of  these  surfaces  are  develop- 
ables.  For  this  it  is  necessary  and  sullicient  that  there  exist  a 
curve  ('In  which  each  ol  the  lines  of  the  surface  are  tangent.  The 
lines  of  the  surface  bein^  determined  by  (  1  ).  (  '1 ),  and  (  '•}  ).  the 
coi'irdi  nates  of  f  are  functions  ot  a.  The  direction  dj".du\dz  of  (' 
iherefore  satisfies  the  equations 


tli/  =  p</:  +  z<lp  +  i/cr, 

where    (ir  =  (    '    +        $(  >i  )  }  »/'r,    and    similar   expressions    hold    for 
\f((      (  p 

I/K.  '/p.  'la~.    (  )n  the  other  hand,  the  direction  of  the  straight  line  (  1  ) 

is  o-ivcn  l»v 

-/./•  =  /•'/-,  '///  =  p<l~., 

so  that    if  the  straight   line   and   curve   are   tangent,  z  must   satisfy 
the  two  eijiiations 


and  thei'efore  we  must  have 

,Jp,!x  -  ,1,;1<T  =  0. 

If  we  replace  i//;  «/x,  <!p.  i/rr  by  their  values,  we  have  as  an  equation 

for  c/H'O  one  which  can   be  reduced   to  the   form 

.!$'•(  'i  )+  /•'<£<  'i  )  -f  ('=  <». 

I-'rom  this  (Mjuation  witli  the  initial  conditions  (  -\  )  we  determine 
two  functions  (f)(n).  Thev  have  been  obtained  as  necessary  con- 
ditions for  the  existence  of  the  developable  surface  through  /,  but 
it  is  not  difficult  to  show  t  hat  if  (/>('<  )  is  thus  determined,  t  he  devel- 
opable sin-face  really  exists,  llciicewe  have  the  theorem: 

'/'/i/'"/'t//i     ilini     Inii'     "t      <l     i-n/ii/  i-lii  ili'i'     i/i)     t  ti'n     i/i  rr/'i^xl/i/i      ,*>i  rtiti'i'X 

fn/'/iii'i/   I*/!   liiii'x   nt    ///c    i-niniriii'iii'i'x. 

Of  course  it  is  not  impossible  that  the  two  surfaces  should  coin- 
cide. but  in  general  they  will  not.  and  we  shall  continue  to  discuss 
the  e.'elicral  case. 

To  the  two  developable  surfaces  through  /  belong  two  curves 
(',  and  i',  the  cuspidal  edsjvs  to  which  the  coii!rniciice  lines  are 


KOt'K    DIMENSIONAL  C  KOMKTRY 
tangent.    The  points  /•'  and  /•' ,  at  which  /  is  tangent  to  C  and  (', 

~  12' 

arc  the  f<><-<il  [mint*  on  /.    The  locus  of  the  focal  points  is  the  focal 

Sll  rtili'i'. 

It  is  obvious  that  any  line  of  the  congruence  is  tangent  to  the 
focal  surface,  for  it  is  tangent  to  the  cuspidal  edge  of  the  devel- 
opable to  which  it  belongs,  and  the  cuspidal  edge  lies  on  the 
focal  surface. 

Let  the  line  /  lie  tangent  to  the  focal  surface  at  /-\  and  Fn,  and 
let  ('  he  the  cuspidal  edge  to  which  /  is  tangent  at  /•'.  Displace  / 
slightly  along  C{  into  the  position  /'  tangent  to  f'}  at  /•','.  The  line 
/'  is  tangent  to  the  focal  surface  again  at  /-'J,  and  the  line  /•'.,/>'.,'  is 
a  chord  of  the  focal  surface.  As  the  point  /•','  approaches  /•',  along 
<\,  the  chord  /''_,/'!!  approaclies  a  tangent  to  the  focal  surface  at  /•',, 
and  the  plane  of  /  and  /'  therefore  approaches  a  tangent  plane  to 
the  focal  surface  at  /•',.  lint  this  plane  is  also  the  osculating  plane 
of  the  curve  ('.  Hence  t/if  oftntlatimj  plmu1  «f  the  ruri'f  c  <tt  /•'  is 
fi/tn/f'/if  tn  tin-  fm'iiJ  siirt'iii'i'  at  l'\t. 

An  interesting  and  important  example  of  a  line  congruence  is 
found  in  the  normal  lines  to  any  surface,  for  the  normal  is  fully 
determined  by  the  two  variables  which  lix  a  point  of  the  surface. 
Through  anv  normal  go  two  developable  surfaces  which  cut  out 
on  the  given  surface  two  curves  which  are  called  litn's  of  I'Hri'iitHr?. 
These  curves  mav  also  be  defined  as  curves  such  that  normals  to 
the  given  surfaces  at  two  consecutive  points  intersect,  for  this  is 
oiilv  one  wav  of  saving  that  the  normals  form  a  developable 
siirtace.  Through  /nti/  ji<n>if  «f  tin'  nurftii'f  ;/<>  tJn'H  t>*'n  fines  >>(' 

i-ll  /•/'<!/  II  /-,'. 

The  t  \\  o  focal  points  on  anv  normal  are  the  centers  of  curvature. 
The  distance  from  the  focal  points  to  the  surface  are  the  principal 
radii  of  curvature,  and  the  focal  surface  is  the  surface  of  centers 
of  curvature.  The  studv  of  these  properties  belongs  properly  to 
the  branch  of  geonictrv  called  differential  geometry  and  lies  out- 
side the  plan  of  this  book.  We  will  mention  without  proof  the 
important  theorem  that  the  lines  of  curvature  are  orthogonal. 

\Ve  shall,  however,  find  room  for  one  more  theorem;  namely, 
that  '/  I'li/ii/ri/i'iiff  '//'  1 1 tit's  //"/•///<//  fa  nil,'  stirt'ni','  is  ii<>nii<il  tn  ///>• 
t'liinili/  ut'  siiri'i/i-i-s  //'///I'll  cut  nti'  ii/i/ii/  ifisfii/n'cs  on  t'l'i'rif  normal 
nii'iisiii'Cil  ti'ii/n  i'iints  ut  t)nf  first  snrtiii'i'. 


LINK  COORDINATES  ;;;;<) 

Let  us  write  tin-  equations  ol  tlu-  normal  in  tin-  lonu 

./•  --  n  +  //•, 

//  -  tf  -f  ////•,  <  4  ) 

z  =  y  -•(-  ///', 

when-  (  a,  /rf,  7  )  is  a  point  of  a  surface  N  :    /.  ///.  n  llic  direct  ion  cosines 
of  the  normal  to  A':   and  /•  the  distance  from  .s'  to  a  point    /'  of  the 

normal.     Then 

/-+  in-  +  tr=  1  : 

\\'helice  I'll  +  IIKJIII  +  if/it  -     0. 

\\'e   have  also  AAt  •+-  n«lrf  -f  /«/y  ~  0, 

since  the  line  is  normal  to  ,s'. 

Suppose,  now,  we  displace  the  normal  slight  lv.  hut  hold  /•constant. 
The  point  /'  goes  into  the  point  (./•  +  '/./'.  //  +  <///,  z  +  dz),  \\here, 
from  (  4  ), 


That  is,  the  displacement  of  /'  takes  place  in  a  direction  normal 
to  the  line  (4).  From  this  it  fnllows  that  the  locus  of  points  at  a 
normal  distance  /•  trom  N  is  another  Mirlace  cutting  each  normal 
orthogonally,  which  is  the  theorem  to  he  proved. 

EXERCISES 

1.  Sho\v  that   the   focal  points  upon  a   line  /  of  a  congruence  can   IT 
(letiiifil  as  the   ]ioint>  at   which  all   ruled  surtaees  which  pav-.  through  /. 
and  arc  composed  of  lines  of  the  congruence,  are  tangent. 

2.  Show  that    the  siii^ular   lines  of  a  i|iiadratic   complex   form   a   rou- 
'4'i'iience,  and  that    the  ^in^ulai1  surface  of  the  complex   is  one   nappe  ot 

the    local    surface   of  the   congruence. 

:i.  Show  that  in  general  there  does  not  exist  a  surface  normal  to  the 
lilies  of  a  congruence,  and  that  the  iieeessarx  and  sullicient  coinlition 
that  such  a  surl'ace  exists  is  that  the  two  developahle  >url'aces  through 
an\  line  ol  the  coii'j;riie!iee  arc  oi't  ho'-onal. 


340  FOUR-DIM EXSIOXAL  GEOMETRY 

4.  Show  that    if   a    ruled   surface   is   composed  of  lines  of  a   linear 
complex,  on  anv  line  of  the  surface  there  are  two  points  at  which  the 
tangent  plane  of  the  surface  is  the  polar  plane  of  the  complex. 

5.  Consider  anv  congruence  of  curves  defined  l>v 

/,(.!',    //,    S,    «,/>)=    0, 

/,(./-,  //,  ,~,  a,  !,}=  0, 

and  deline  as  surfaces  of  the  congruence  surfaces  formed  l>v  collecting 
tin*  congruence  curves  into  surfaces  according  to  anv  law.  Show  that 
on  anv  congruence  curve  ( '  there  i-xists  a  certain  number  of  focal  points 
such  that  all  surfaces  of  the  congruence  which  contain  ('are  tangent 
at  these  points. 

6.  Trove  that  it  the  curves  in  Ex.  5  are  so  assembled  as  to  have  an 
envelope,  the  envelope  is  composed  of  focal  points. 


CHAPTER   XVIII 

SPHERE  COORDINATES 

146.  Elementary  sphere  coordinates.  Another  simple  example  of 
;i  geometric  iigure  determined  1>\  four  parameters  is  the  sphere. 
\Ye  mav  take  the  quantities  </,  i',/",  /•„  which  ti\  the  center  and 
radius  of  the  sphere 

( .r  -  -/  )-  +  (//  -  <•)-  +  {2  -/)'J  =  r,  (  1  ) 

as  the  ei  lordinates  of  the  sphere,  and  obtain  a  four-dimensional 
uvoinetrv  in  which  the  sphere  is  the  element. 

It  is  more  convenient,  however,  to  use  the  pentaspherieal  coor- 
dinates .r  of  a  point  and  take  the  ratios  of  the  eoet'licients  "  in 

the  eiiiiation 

" i'''i  +  Va  +  '  Vs  +  '  V-4  +  V; =  (-  ) 

ot  a  sphere  as  the  sphere  coordinates.  This  is  essentially  the  same 
as  taking  «/,  <\  _/',  and  /•.  In  fact,  if  .r,  are  the  coordinates  of  sj  1 1  7, 
then  by  (4),  Jj  117,  t'<iiuition  (-)  can  be  written 

( rt,  +  i<t. )  ( .r  -f  //-  +  r )  +  L'  </.,./•  +  -2  a. ,ii  +  -2  ,/ (,r  -  ((^  —/</.)=  0,    ( ;i ) 

and  the  connection  with  (  1  )  is  obvious. 

By  J;  111*  two  spheres  are  orthogonal  \\heii  and  only  when 

"A  +  "•/',+ "./',+  "/;4  +  "A-  °'  ( 4 } 

the  coordinates  .r   IKMUJT  assumed  orthogonal. 
Consider  now  any  linear  equation 


where  <\  arc  constants  and   nt  sphere  coordinates.     It  we  deter 
a  sphere  with  coi'irdinates  <•.,  (  f> )  is  the  same  as  (  I),     llciice 

.1    Ititiii/'   ,'ijiiiit t'in    ni    i'lfitit'uftti'i/   xfifit'n1    t'nu'i'tlttitift'x    /'i  j>/'i  *t  ///.•>•  <i 

i'"/ii[il>.i-  ,it  >y///,  /VN  i-'iHX/xt/ili/  "/'  >'/'//'  /•'  N  npthtii/iiHitl  f"  <<  ti.i'il  Xj>/t>'l't'. 
If  tl,,-  fi.r,  ,1  .sy, //,/-,  ,'*  Kftn-tK/  tin  ,;,„//>/,'./•  rt'tixfxtx  -/  >y//>  r,  x  t/ir<>t<://< 
th,  .;///,,•  itf'  f/u'  xi >,,•/,//  K/i/ifi-i'  <!//</  />•  ril//i-il  'i  xix-fifll  t'»nn>h'X. 


.'Mli  FOIK    DI.MKNSIONAL   ( ',  K(  >.M  KTRY 

The  word  "complex  "  is  used  in  tin-  same  sense  ;is  in  £  1 1  •'>,  for 
it  'i  ,  /^  ,  7,  c\  arc  lour  spheres  which  satistV  (4),  uny  sphere1  wliieli 
siitislies  (  {)  has  the  coordinates 

<j,  -f  X^-f  M7,  -f  "£,- 
('niisider     now     the     two     simultaneous     equations     in     sphere 


Spheres  which  satisfv  both  of  these  equations  belong  to  two 
complexes.  Thereloiv  //'•<<  ximii/tdHi'tiux  ////><//•  i'<jn<itn>n8  in  i'l>'//n'n- 
tiini  ffi/ti'/'i'  ctiti/'iltnxtfH  <>/'i'  xxtixfifd  /<//  nji/ti-i'i'x  ichic/i  <ir<'  nrthuf/nnul 

t"  dm  //./•/'(/  ,sy<//c/vx.  These  spheres  form  a  bundle,  for  if  a i%  /^,  yt 
are  anv  three  spheres  which  satisfy  (<>),  any  sphere  satisfying  (_!>  ) 
has  t  he  coi'irdi nates  n(  -)-  X/d,  -)-  /xy,. 

All  spheres  which  belong  to  the  two  complexes  in  (<i)  lu-loiig 
to  the  complex  ^vl'V/,d~  X^, '/,-",—  (h  mid  anv  t\\'o  complexes  of  the 
latter  form  determine  the  bundle.  Among  these  complexes  there 
arc  in  general  two  and  onlv  two  special  ones,  and  so  we  reach 
attain  the  conclusion  that  a  lnnnllt'  ol  spheres  consists  in  general 
ot  spheres  through  two  fixed  points. 

Three   linear  equations, 

determine  spheres  which  are  orthogonal  to  three  base  spheres. 
'1  lie>e  spheres  form  a  pencil,  since  it  n  and  /^  are  anv  two  spheres 
satisfying  (7),  anv  >phcre  which  satisfies  (7)  has  the  coordinates 

\\Y  shall  not  proceed  further  with  the  studv  of  the  elcinciitarv 
coordinates,  as  more  interest  attaches  to  the  higher  coordinates, 

defined    111     the    next     >eet  loll. 

EXERCISES 

1.  Consuler  i]M.  ipiailratic  complex  y//^.//^^.  =  0.  (ti^.  —  ti^.}  and 
•  i-  polar  linear  complex  of  a  sphere  r(,  del'med  h\  the  equation 
^  ",'',"/  <  I.  I  t  1  In-  del  erm  ina  nt  '/,;  ---  0,  sho\\  t  hat  t  o  a  n\  sphere  (^ 
c"rres]ionds  one  polar  complex,  and  conversely. 

L'.  Show  that  if  i- 1  lies  in  the  polar  complex  of  //•  .  then  n\  lies  in 
the  polar  complex  of  /•  .  The  t  \vo  ^pliero  /•  and  n\  are  >aid  to  he 


SPHKKK  COOKDINATKS  ;54;} 

3.  Show  that  tin-  pencil  tit'  spheres  defined  liy  two  conjugate  spheres 
lias    in    common    with    the    quadratic    complex    two   spheres    which    are 
harmonic  conjugates   of   the    tirM    two  spheres   (tlie   cross   ratio  of   four 
spheres   of  a    pencil    is   defined   as   in   the   case   of   pencils   of   planes). 

4.  Show  that  the  assemblage  of  all  special  spheres  forms  a  quadratic 
complex.    Show  that    anv    two  orthogonal   spheres   are   conjugate   with 
roped  to  this  complex,  and  that  the  polar  complex  oi  anv  sphere  r   is 
the  complex  ot  spheres  orthogonal  to  i\. 

5.  Show  that   the    planes   which    belong  to  a   quadratic   complex   en- 
velop a  quadric  surface. 

6.  Show  that   anv  arbitrary  pencil  of  spheres  contains  two  spheres 
which  belong  to  a  Lfiveii  quadratic  complex,  and  that  anv  arbitrary  point 
is  the  center  of  two  spheres  of  the  complex. 

7.  Show   that    the    locus    of   the    centers    of   the    point    spheres    of   a 
complex    with    nonvaiiishin^  discriminant-   is   a   eyelide. 

8.  l>etine  as  a  *//»/////  .sy^r/W  complex  one  for  which  the  discriminant 

a  t>.  vanishes  but  so  that  all  its  first  minors  do  not  vanish.  Show  that 
such  a  complex  contains  one  singular  sphere  which  is  conjugate  to  all 
spheres  in  space.  Show  that  the  complex  contains  all  spheres  of  the 
pencil  determined  by  the  singular  sphere  and  anv  other  sphere  of 
the  complex,  and  that  all  spheres  of  such  a  pencil  have  the  same  polar 
complex. 

147.  Higher    sphere    coordinates.     Let    .t\    be    orthogonal    penta- 
spherical  coordinates  whereby 

Gj(j-)=V/j=0      and      j;(,/)=V<r,  (1) 


be  the  equation  u|'  a  sphere.    To  the  live  quantities  ti  ,  a.,,  (/3,  a^  (/,. 
\\'e  will  adoin  a  sixth  one.  <i  ,  defined   by  the  relation 


and  the  ratios  of  these  quantities  are  taken  as  the  coordinates  of  the 
sphere.  This  is  jiistitied  by  the  fact  that  if  the  sphere  is  given, 
the  coordinates  are  determined:  and  it  the  coordinates  are  given, 
the  sphere  is  determined. 


344  FOUR-DIMENSIONAL  GEOMETRY 

More  generally,  if  a(,  a  t,  a.{,  ^4,  <r.,  afi  are  six  quantities  such  tluit 


with  the  condition  that   the  determinant    a,,.    shall  not  vanish,  the 
ratios  <it  :  a^.  may  he  used  as  the  coordinates  of  the  sphere.     Equa- 
tion   (4)   then  goes    into  a  more  general   quadratic   relation.     We 
shall,   however,   eontine   ourselves  to  the  simpler  at. 
By  (^-0),  §  llM,  the  radius  of  the  sphere 


is 


Consequently,  to  change  the  sign  of  a^  is  to  change  the  sign  of  the 
radius  of  the  corresponding  sphere.  If,  then,  we  desire  to  maintain 
a  one-to-one  relation  between  a  sphere  and  its  coordinates,  we  must 
adopt  some  convention  as  to  the  meaning  of  a  negative  radius. 
This  we  shall  do  bv  considering  a  sphere  witli  a  positive  radius  as 
bounding  that  portion  of  space  which  contains  its  center,  and  a 
sphere  with  negative  radius  as  bounding  the  exterior  portion  of 
space.  Otherwise  expressed,  the  positive  radius  goes  with  the  inner 
surface  of  the  sphere,  the  negative  radius  with  the  outer  surface. 
A  sphere  with  its  radius  thus  determined  is  an  vrtfnt>'<1  .v////c/v. 

If  the  sphere  becomes  a  plane  the  positive  value  of  </  is  associ- 
ated with  one  side  of  the  plane,  the  negative  value  with  the  other. 

A  sphere  is  special  when  and  only  when  at.  —  0. 

148.  Angle  between  spheres.  By  ^119  the  angle  between  two 
spheres  with  coordinates  <it  and  /<_  is  defined  by  the  equation 

,t  I,  -f-  ,i  I,  4-  ,/  /,  +,,/,+  l(  /, 

-* 


cos 

a  ft 

>j    C 

Ilt-nce  the  angle  0  is  determined  without  ambiguity  when  the 
signs  of  the  radii  of  the  two  spheres  are  known.  If  both  radii  are 
positive,  tf  is  the  an^le  interior  to  both  spheres;  if  both  radii  an- 
negative,  0  is  exterior  to  both  spheres  :  and  if  the  radii  are  of  opposite 
>i'_,rn.  6  is  interior  to  one  sphere  and  exterior  to  the  other. 

For  special  spheres  the  angle  defined  bv  (1  )  becomes  indeter- 
minate. More  precisely,  it  <i  is  a  special  sphere  the  coordinate 


SIMIKKK  COORDINATES  :)45 

^=0  and  the  other  live  sphere  coordinates  are  the  pentaspherieal 

coordinates   of   the   center  of   the  sphere.    Then-lore   the   condition 
that    the   center  of   the   special  sphere  ni  lie  on   another  sphere  /-,  is 

"/'!+•  "-A+  "A+  "A+  "A=  °- 

Therefore  if  iii  is  a  special  sphere,  /<,  any  other  sphere,  and 
0  the  an^le  between  <r  and  />.,  cos  #  is  infinite  when  the  center  of 
</;  does  not  lie  on  /<(,  lint  is  ||  when  the  center  of  </(  lies  on  //(. 

.1    NJH'<'i'fll    XJlJllTt'    thf'I't'fvl'C    UKl/CCH    <t  It  I/    illl'/Ii'    With     it    HJiJld'C    "II    tr/t<<-/t 

it*  r,-///,r  //<•*. 

\Vhell    #  -  :(  'lie  +1  )^,    >;(</,/-)=  f/  //,-}-  </A+  "./'.+  "/',+  "A=  °- 

and   eonverseh".      llciice  we  inav  sav  : 


When  6  =  (I,  |(  fi,  /-)=  "/',-(-  "./'.,+  "./';+  "/'.,+  "-/'•,+  ",/V  °<  all<1 
eonverselv.  In  this  case  the  spheres  are  said  to  lie  tangent,  but  it 
is  to  he  noticed  that  spheres  are  not  tangent  when  0  =  TT.  The  dif- 
ference between  the  cast's  in  which  0  =  0  and  those  in  which  6  =  TT 
lies  in  the  relation  to  each  other  of  the  space  which  the  spheres 
bound.  In  fact,  if  two  spheres  which  are  tangent  in  the  elementary 
sense  lie  outside  of  each  other,  they  are  tangent  in  the  present 
sense  only  when  one  is  the  boundary  of  its  interior  space,  and  the 
oilier  is  the  boundary  of  its  exterior  space:  that  is,  the  two  radii 
have  opposite  sij^ns.  If  two  eleinentarv  spheres  are  tangent  so  that 
one  lies  inside  the  other,  thev  are  tangent  when  oriented  only  it 
the  radii  have  the  same  si<j'n.  We  say: 


Two  planes  are  tangent    when  they  arc  parallel  or  intersect    in  a 

i  •          i  *       . .    .-  .  -  i 
nun  nun  in  line  (  I1,  \.  r»,  ^  M  ). 

It    is  ob\  ions   that    all   the.se   theorems   are   unaltered    b\    the   use 
of  the   more  genera!   sphere  coordinates  of  £  1 -J  1 . 

'1  lie   aii^'lc   Ht    made   by    I  he   sphere   <ti   with    the   eoi'ird  mat  e   sphere 
.^  =-:  0   is  L;M yen   by  t  he  c(|uat  ion 

>*0,-.     -">• 


:l4fi  Fol'i;   DIMENSIONAL  GEOMETRY 

Consequently  we  have  the  theorem: 

/>//  tin  us,'  nf  ort}ii'ij"inil  i-nnfil  i  null's  .iv  iiml  t//f  xpln'r?  I'nfinlintiti'x  <t:, 
tin1  til',1  <'"i'iril  i  mil,  'x  it  ,  <?,.  n  .,  it  ,  ii.  nj  tin  if  xpln't'i'  iii'f  propnrtimntl  f<>  f/n- 
i-iisi/ii'x  at'  tin-  ittiij/i'x  /r///i'/i  f/nit  sphere  nntkt'x  /<•////  tin1  I'tH'h'iUnittc 

sr/,,r,s.  ' 

149.  The  linear  complex  of  oriented  spheres.  Equation  (1)  of 
vj  14s  m;tv  be  written 

a  I,  +(//-   -f  ,/  I  4-  </  /*  +  <tj>.  +  (/  /•    cos  0  =  0.  (  1  ) 

11  -    «  ;)    <i  41  o    u  u    tj  v      * 

Ciuisitler  no\v  u  liin'iir  eijuation 

(\"i  +  '  V'-j  +  '  V';(  +  'V,  +  'V';,  +  '',;"„  =  °'  (  L>  > 

where  //,  arc  higher  sphere  eotu'dinates  and  r(  ai't-  constants.  The 
spheres  \\hirh  satisfy  this  equation  form  a  fint'<u'  rompli'j'. 

This  tMjiiation  may  in  genend  be  identified  with  (1)  hv  deter- 
mining a  lixcd  sphere,  called  the  /«/.sr  x/>/n'/-<',  \\ith  the  coordinates 

,/.=  ,-.,  (i'  =  l,   2,  3,  4,  f>),     ",=  /\7'V  +'•;+'-;+,•;+,-;,         (  :{  ) 
and  determining  an  angle  0  l»v  the  etjiuition 

a  f  cos  tf  —  <;.  (4  ) 

Equation  ('!)  is  then  satisfied  hv  all  spheres  which  make  the 
angle  0  \\ith  the  base  sphere.  This  angle  is  equal  to  <>  when  and 
only  when  CH=//(,  :  that  is,  wlien  %(<•)—().  In  the  latter  case  the 
complex  is  called  */><•<•/,//. 

We  put   these  results  in  the  form  of  the  theorem: 

,1  //iii-iir  I'm/I  j>lt'.r  rnitsist*  in  i/t'ihfii!  i't'  xpht'i'ex  flitting  a  fi.i't'il 
xphi're  uinltT  <i  ,'nfisfin/f  /nnjli'.  It  j~  (<•)=()  tlnj  cuniph'jc  is  special 

<I/I<1    ('Hltxixtti    nt'   Xjl}nT<'}<    fillt</i'Ht     fn    ll     fi.l'<'</    Xphl'l't'. 

The  \\'ords  "in  general"  have  been  introduced  into  the  theorem 
because  ot  the  exceptional  cases  which  arise  when  the  base  sphere 
is  special  :  that  is.  when  ii,  -  ".  In  that  case  the  an^'le  0  cannot  be 
det  ermined  t  ]•<  mi  (  4  ). 

If  at  the  same  time  that  <(f  —  0  the  complex  is  special,  then  <•  .  =  (.>, 
and  the  complex  is 


with  "^  i-r       (I.    'I  hen  <•    are  the  coordinates  of  a     oint,  the  center  of 


S1MIKKK   COOKDI. \.\TKS  :J.J7 

If  when  nf  =  0  the  complex  is  not  special,  then  c  /-(I,  and  the 
anj^le  61  cannot  be  determined.  A  particular  rase  in  which  this  mav 
happen  is  when  <•  —  '• .,  —  <• .---  <•  =  f.  =  0,  and  the  complex  is 

",       °- 
This    equation    is    satisfied    bv    all    special    spheres.     Therefore    nil 


There  remain  still  other  cases  in  which  <it.  0,  but  '•  =£  '>.  The 
base  sphere  is  then  special  and  the  anisic  0  is  iniinite,  but  the  com- 
plete definition  of  the  complex  is  through  its  equation. 

EXERCISES 

1.  Prove   that    the   base   sphere    of   a    complex    is    the    locus    of   the 
centers   of   the  special   spheres   which   belong   to   the   complex. 

2.  Prove  that  if  r  =  0  in   the  equation  of  a  complex,  the   complex 
consists  of  spheres  orthogonal   to  a    fixed  .sphere,  as   in    j  1  1<>. 

3.  Prove  that    in  a  special  complex  the  coefficients  in  the  equation 
of  the  complex  are  the  coordinates  of  the  base  sphere. 

4.  Prove  that"  all   planes  together  make  a  special  complex  with  the 
base  sphere  the  locus  at   intinitv. 

5.  Show  that   all  spheres  with  a  fixed  radius  form  a  linear  complex 
and  del erm me  the  base  sj there. 

G.    Piscuss    the    relation    between    two   complexes    whose    equations 
differ   onlv    in    the   sign    of  the   last    term. 

1 .    Two  linear  complexes   >  ?•,//,.=  0  and  'v'/,-"1  —  ' 


!'.  Show  that  the  complex  consisting  of  spheres  ort  hopnial  to  a 
nonspecial  luise  sphere  is  in  involution  \\ith  the  complex  of  all  special 
spheres. 

10.    Show  that   the  six  complexes  t/t  -     0  are  pair  bv  pair  in  involution 

and  determine  the  relations  of  the  base  spheres. 


rns  Fom  DIMKNSIONAL  (JKOMKTUY 

11.  ( 'n>iji/i/nfi'  spheres  with  respect  to  a  linear  complex  are  such  that 
anv  sphere  tangent  to  both  belongs  to  the  complex,  and  anv  sphere  of 
the  complex  tangent  to  out'  is  tangent  to  the  other. 

Show  that  if  i\  is  anv  sphere,  the  conjugate  sphere  has  the  coordinates 


"V  ,.-'      '' 

12.  If  a  com] ilex  is  composed  of  spheres  orthogonal  to  a  base  sphere, 
show  that  the  conjugate  of  a  sphere  >'  is  the   inverse  of  >'  with  respect 
to  the  base  sphere. 

13.  Find    without    calculation    and   verifv    hv   the    formulas  the   con- 
jugate  of  a  sphere   with    reference   to  a   complex   of  spheres  with   fixed 
radius  //. 

14.  Show  that  the  conjugate  of  a  sphere  with  respect  to  the  complex 
of  special  spheres  is  the  same  sphere  with  the  sign  of  the  radius  changed. 

150.  Linear  congruence  of  oriented  spheres.    The  spheres  common 
to  two  linear  complexes 


form  a  x}>h<>ri'  >'n)i</ri/i'i/i'i'.     Anv  sphere   of  the   congruence  (1  )  al 
belongs  to  anv  complex  ot   the  torm 


and    anv   two    complexes    ot     iorm   (  -  )    can    be    used    to   defin 
coiio-Tnelice. 

Now     (  '_}  )    represents    a    special     complex     when    A.    satisfies    the 

eiinat  ion 

^  (  <i  +  \f-  )      <)  : 

that   is,  £(,i)  +  -2  \^(it,  /-)  +  \'-'£(  /•)  .-n.  (:'») 

Hence.  ///  i/i  in  1'i'L  it  .v////,'/v  coinjrih  )!'•>'  r-///x/x/.v  »('  .sy»//(  /•/  x  lit/n/fiif 
fu  firn  xjilii'l'i'X,  fill/I'll  lli/'t'rf  /•/'.!'  XJi/HTCX. 

The  exceptional  cases  occur  \\'lien  the  roots  of  equation  ('•'>) 
are  either  illusive  or  eipial.  In  the  first  case  equation  (  •!  )  is 
identically  satisfied  and  all  complexes  of  (  •_'  )  are  special.  The 
congruence  niav  then  be  defined  in  an  infinite  number  of  \\avs 
as  composed  of  spheres  tangent  to  t  \\  o  dii'ectrix  spheres.  The 
condition  that  (  '-\  )  be  ident  icallv  satisfieil  is  f  ('')".  £(/')-~0, 


SIM  IKK  K  COORDINATES  340 

£(<>,  //)=:<).  The  first  t\vn  ('((tuitions  suv  tliat  the  defining  com- 
plexes lire  special  ;  the  third  equation  says  that  the  base  sphere 
of  either  lies  on  the  other. 

If  the  two  roots  of  ('•'>)  are  equal,  there  is  onlv  one  special  com- 
plex; in  the  pencil  (-).  Suppose  we  take  this  as  N  ,/  ti  =  0.  Then, 
since  the  roots  of  (  o )  are  equal,  £  (  ".  /<)  =  <>.  '1'his  says  that 
the  hast1  sphere  of  the  special  complex  belongs  to  the  complex 
V/,iMi=<). 

151.  Linear  series  of  oriented  spheres.  Consider  now  the  spheres 
common  to  the  three  complexes 

2'V/,.=  0,      2/M/,=  0,      V,v/.=  0,  (1) 

which  do  not  define  the  same  congruence.  These  spheres  form  a 
thii'iir  xtT/fx. 

A   sphere  of  the  series  (1  )  belongs  also  to  any  complex  of  the 

2(Xrt,+  /iAf+i"Y)Mi=0»  (-) 

and  any  three  linearly  independent  complexes  (-)  may  be  used  to 
define  the  series.  Among  the  complexes  (-!)  there  are  a  simply 
infinite  set  of  special  complexes:  namely,  those  for  which  X,  ft.  and 

v  satisfy  the  equation      <.    .  ,  ., 

f  (Xrt  +  /JL(>  +  vc )  =  0.  ( 3 ) 

Tin1  xjiht'ri'ti  <it'  t/n-  x/'/vVx  ( 1  )  f<>rm,  therefore,  n  nne-rfimennwHal 
<\rfi')if  "f  xphi'n'K  ii'Jiii'h  <tr>'  f<ni</i'>it  /<>  a  onf-dimenxiontfl  i'.rf>'tif  <>f 
ilirt'ftri.r  spheres. 

The  nature  of  the  series  depends  on  the  character  of  equal  ion  (  o  ). 

We  shall  assume  that  the  discriminant  of  (o)  does  not  vanish. 
I  f  the  quantities  (  X,  /LI,  v)  are  for  a  moment  interpreted  as  trilinear 
point  coordinates  in  a  plane,  equation  ( '-\ }  will  represent  a  conic 
without  singular  points;  hence  it  is  possible  to  find  three  sets  of 
values  which  satisfy  ( :> )  and  are  linearlv  independent.  We  have 
corresponding  to  these  values  of  (A.,  /u.  v)  t  hree  linearlv  independent 
special  complexes,  and  mav  assume  without  loss  of  generality  that 
they  are  the  three  complexes  in  equations  (I  ). 

Then  any  one  of  the  directrix  spheres  has  the  coordinates 
(  $  I  I'.') 


:;:>()  Koru  DIMENSIONAL  GEOMETRY 

Now  if  ft,,  /tf,  and  7,  are  any  three  spheres  of  the  series  ( 1  ),  it  is 
obvious  that  the  spheres  >\  in  (4)  satisfy  the  three  equations 

V, ,,...,().       Vjtfr   =-:(>.       V^y;—  0.  (li) 

— '    '  ^-r    ;  A1  '' 

Conversely,  any  sphere  satisfying  equations  (»!)  satisfy  (4),  for 
three  solutions  ot  (li)  are  rr,  /<(.,  <•,,  and  the  most  general  solution 
is  therefore  X<it  -(- /-<A  +  i"',.  where  (since  rr  are  sphere  coordinates) 
equation  (  o )  must  be  satisfied. 

Hence  tin'  ilirt'i'tn.r  xphrrfft  form  nnntln'r  /hh'/ir  .sr/vVx. 

The  special  complexes  which  may  dcline  the  series  (<i)  are 

where  %(p<1,  +  afi,  +  T7, )  =  "• 

The  base  spheres  of  these  arc  simply  the  solutions  of  (  1  ).  Hence 
tin-  i liri'i't /•/'.>•  Hjtheri'it  »f  tin1  xi-rit'x  (li)  itn-  tin-  x/>/n'r>'x  <>f  (  1  ). 

}\'<'  fi<i>'<\  t/ii'irt'oft',  (fro  xi'rii'x  <>f  xf>/it'i-i'x  xit<-fi  tlnit  nif/i  xf>hfn'  "f 
nn<'  xt'rifx  ix  tin'  fioii/fnt  t<>  <'<t<-li  xf>1n'r>'  <f  tin-  "tin  r. 

(hi  flu'  <>f /iff  l«Di<l<  //"  tiro  xji/it'ri'x  if  tin'  mi/i/i'  xi'n'i'x  nr<'  (ittii/i'uf. 
To  proye  this  note  that  by  ( f> )  we  have 

XfJLJ;  (  ".    I  )  +  fJLl'j;  (  /..    f  )  +  1>X£  (  f,    ,1  )  -  0, 

and   no  one  of   these  coefficients  can   vanish   under  the   hypothesis 
that  the  discriminant   ot  (o)  does  not  vanish.     But  </,,  /»,-.  '',  are  any 
three  directrix   spheres,  and   hence  the  theorem. 
By  sj  1  1  •")  we  are  able  to  say   immediately: 

///  ///<•  i/i'th'/'ii/  i'iixi'  fin-  xjt/tcrt'x  'if'  n  Inii'iir  xi'n'i-x  r///v/<y>  <t 


We  shall  not.  discuss  the  special  forms  of  the  linear  series  arising 
when  the  discriminant  of  equation  ( •'> )  vanishes. 

152.  Pencils   and   bundles   of  tangent   spheres.     It    «{  and   />,   are 

any  two  spheres,  then  ,.  , 

pi/ 1  =  a :  -\-  \fi.  (  1  ) 

is  a  sphere  when  and  only  when  N//7*|  =  0;  that  is,  \\heii  n:  and 
A  are  tangent.  In  this  case  (1  )  represents  ~/^  spheres,  each  of 
which  is  tangent  to  each  of  the  others.  \Ve  call  this  a  jn-ni-il 
"t  fiiiti/i  at  xnlitTi'K.  In  the  notation  ot  ^117  the  condition  for  a 
special  sphere  in  the  pencil  is 

r/fi+U=0,  C2) 


SI'HKKK  COOUDINATKS  ;;:,] 

so  that  there  is  onlv  one  special  sphere  in   the  pencil  unless  <i    and 
/',,  ami  consequently  all  spin-res  of  the  pencil,  arc  special. 
The  condition   for  a  plane   in   the  pencil   is 

^4- '",+  X('',+  #'6)  =  (l«  (•») 

so  that  there  is  onlv  one  plane  in  the  pencil  unless  all  the  spheres 
of  the  pencil,  including  </.  and  /<(,  arc  planes. 

In  general  the  special  sphere  and  the  plane  are  distinct  from 
each  other.  Therefore  the  special  sphere  is  a  point  sphere  \vhose 
center  is  in  finite  space.  This  center  lies  on  all  spheres  of  the 
pencil  l>v  vj  14S.  Hence  the  pencil  is  composed  of  spheres  tangent 
to  each  other  at  the  same  point.  Such  spheres  have  in  common 
t  \vo  minimum  lines  determined  hv  the  intersection  of  the  point 
sphere  and  the  plane  of  the  pencil.  These  statements  mav  l»e  veri- 
iied  iiniilvtieullv  hv  writing  the  equations  of  the  spheres  in  the 

ii.i*!  I 

form    (  o  ),    £111. 

Special  forms  of  a  tangent  pencil  mav  arise,  however.  For 
example,  it  mav  consist  of  spheres  having  two  parallel  minimum 

I  fc.  I  ii  I 

lines  in  common.  The  special  sphere  and  the  plane  in  the  pencil 
then  coincide  with  the  minimum  plane  determined  l>v  these  mini- 
mum lines.  Again,  the  pencil  mav  consist  of  point  spheres  whose 

centers   lie   on    a   minimum    line.     The   plane   in    the   pencil    is  then 

i  i 

the  minimum  plane  through  that  line.  (  )r  the  pencil  mav  consist  ot 
parallel  planes  (  £  -IS  ).  The  special  sphere  in  the  pencil  is  then  the 
plane  at  infinity  unless  all  the  planes  ot  the  pencil  are  minimum 
planes  and  therefore  special  spheres.  Fimillv,  the  pencil  mav 
consist  of  planes  intersecting  in  the  same  minimum  line  (  £  Ts  ). 
The  special  sphere  is  then  the  minimum  plane  through  that  line. 
It  ",  /«,,  and  c:  are  three  spheres  not  in  the  same  pencil,  then 

px,—  »,+  X/-(  -f  fj.'-,  (  1  ) 

is  a  sphere  when  and  onlv  when  the  three  split-res  are  tangent  each 
to  each.  In  that  case  eijiiation  (  1  )  defines  s  ~  spheres,  cadi  o|  which 
is  tangent  to  each  of  the  others.  It  is  a  linnll,  <</'  /</////.///  x/»//r>vx. 
'1  here  ;ire  in  the  bun  die  /.  '  speeial  spheres  determined  l>v  I  he  ei  |  uat  ton 

ii    |    \/>:  -f  /u'^      <>,  (  •">  ) 

and  ~f.1  [daiies  determined  l>v  the  etpiation 

",+  i«f+  \<f>,  +  if>.  )  f  M('\+  /'•  )-    (>-  ( (;  ) 


3"> 2  FOTK  DIMENSIONAL  GEOMETRY 

In  general,  equations  (  "> )  and  ( i> )  have  only  one  common  solution, 
so  that  the  special  spheres  are  point  spheres.  Since  all  spheres  of 
the  bundle  are  tangent,  the  centers  of  the  point  spheres  lie  on  a 
minimum  line  which  lies  on  all  the  spheres  of  the  bundle.  The  point 
spheres  and  the  planes  form  each  a  pencil  in  the  sense  already  dis- 
cussed, so  that  any  point  of  the  common  minimum  line  is  the  center 
of  a  point  sphere  of  the  bundle,  and  any  plane  through  the  minimum 
line  is  a  plane  of  the  bundle.  From  that  we  may  show  that  any 
sphere  which  contains  that  minimum  line  and  is  properly  oriented 
belongs  to  the  bundle.  For  let  >\  be  such  a  sphere  and  <t';  any  plane 
of  the  bundle.  Since  >•.  and  n\  have  one  minimum  line  in  common, 
they  have  another  minimum  line  in  common  which  intersects  the 
first  one  at  a  point  /'.  Let  //  be  the  point  sphere  with  center  /'. 
Then  /-,  is  tangent  both  to  a',  and  1>\  at  /',  and  therefore 

if    the    proper    sign    is    given    to    «'r      Hut    </'—  n,  -f-  X'^-f-  ftV,    and 
//=  ff.-f  X'^+^'V,,  so  that 

whence   rt  belongs  to  the  bundle. 

Summing  up,  we  sav  :  ///  </<')ii'i'n/  <i  l»niiUi>  <>f  (tinifenf  spheres  <*nn- 
,\v,v/.s>  /if  iill  tin'  ~jz "  xpln'WR  H'hifh  have  <t  nnnmuiin  l/iif  1/1  i'n/>uni>n 
<ni<l  >if  nn  other  xphrrex. 

To  avoid  misunderstanding  the  student  should  remember  that 
we  are  dealing  with  oriented  spheres  and  that,  for  example,  three 
elementary  tangent  spheres  which  lie  so  that  two  of  them  are  tan- 
gent internally  to  the  third,  but  externally  to  each  other,  cannot 
be  so  oriented  as  to  be  tangent  in  the  sense  in  which  we  now  use 
the  word. 

Special  forms  of  bundles  deserve  sonic  mention.  In  the  lirst 
place,  we  notice  that  not  all  the  spheres  can  be  point  spheres  :  sine*1, 
if  they  were,  the  centers  of  three  spheres  would  be  finite  points 
not  in  the  same  line  but  in  the  same  plane,  so  that  each  is  con- 
nected with  the  other  by  a  minimum  line,  which  is  impossible. 

The  spheres  of  the  bundle  may.  however,  all  be  planes.  Then 
the  special  spheres  must  be  minimum  planes,  which,  since  they  are 
tangent.  must  form  a  pencil  of  minimum  planes  tangent  to  the 
circle  at  infinity  at  the  same  point  (  vj  4*  ).  All  planes  of  the  bundle 


SIMIKKK  COOKDINATKS 


must    pass    through    this    point,    and    it    is    evident    that    anv    two 

planes  throutrh  this  point  either  intersect   in  a  minimum  line  or  are 
i  n  » 

parallel,  and  in  each  case   are   tangent.     Hence.  </.«,•  /i   .vy/»-/-/<//  <•</.•<.•  <i 

lillllilli'    at     tillH/i'llt    XltJll't'l'X    IIHII/    I'niisist     nt     X  "   Jit'lHi'X    f/l/'"»i//l     f/lf'     $illH'' 

jaunt  n/i  tin'  / //nii/i //it  ri/  ri/'i'/f  nt  infinity. 

153.  Quadratic  complex  of  oriented  spheres.    Consider  the  quad- 
ratic complex  defined   by  the  equation 

V,.(,r.-=0.  (1, 

This  is  the  form  to  which   in  general  an  equation   <>f  the  second 
degree  in  j\  can    be   reduced,  and   we  shall  consider  only  this  case. 
Since  the  sphere  coordinates  satisfy  the  equation 

V;r=0,  (-2) 

±-i    ' 

the  same  complex  (1)  is  represented  by  any  equation  of  the  form 


Now  let  //,  he  a  sphere  of  (  '•}  ),  and  zt  anv  sphere  tangent  to  //,, 
and  consider  the  pencil  of  tangent  spheres 

pn.=  yi+'tei.  (4) 

This  pencil  has  in  common  with  ('•}')  the  two  spheres  corre- 
sponding to  the  values  of  X  obtained  bv  substituting  from  (  4  )  in  ('•>). 
This  gives,  with  reference  to  the  fact  that  //,  satisfies  (:^), 


The  one  common  sphei'e  is,  then,  alwavs  i/  .  as  it  should  be.  but 
the  other  is  in  general  distinct  from  //  and  coincides  with  it  when 
and  otilv  when  zt  satisfies  the  relation 

V    '•  ?=  (>  : 


(  ,»  ) 

Thi>  complex    is   called   a   fiini/rnt   ////»•///•  .•<,//////,  ./•. 

l-'rom  the  derivation  a  t  an^eiit  linear  complex  through  a  spliere  i/i 

is  a  linear  complex  which  contains  //   and  ha:-  the  property  that   any 
jicncil   ot    tangent   spheres  Itelon^'in^   to  the   linear  complex    which 


;;:»4  rnri;  OIMKNSIONAI.  CKOMETKY 

contains  //,  has.  in  common  with  the  quadratic1  complex,  onlv  the 
sphere  //  doubly  reckoned,  unless  the  pencil  lies  entirely  in  the 
quadratic  complex. 

This  detinition  is  analogous  to  that  given  in  point  space  for  a 
tangent  plane  to  a  surtace  by  means  ot  coincident  points  of  inter- 
section dt  a  line  in  the  tangent  plane.  The  exceptional  cases  of 
pencils  entirely  on  the  complex  are  analogous  to  tangent  lines 
which  lie  entirely  on  the  surface. 

It  mav  also  he  noted  that  if  //,+  '///,  is  any  sphere  of  (  1  )  adja- 
cent to  //(.  so  that  "N  '•,//•'///,  =  ()  and,  from  (  '2  ).  "N  //,<///,  =  ".  1  he  sphere 
lies  also  in  (  •">  ).  The  tangent  linear  complex  contains  all  spheres 
of  the  quadratic  complex  adjacent  to  i/^ 

Since  /j.  is  arhitrary  in  (  ;">  )  the  quadratic  complex  (  1  )  has  a  pencil 
of  tangent  linear  complexes  through  any  sphere  //_.  Among  these 
there  is  in  general  one  and  only  one  which  is  a  special  complex, 
for  the  condition  that  (  ;>  )  he  special  is 

]£(,;+  /i  )-//;-•  ^  0, 

which,  if  we  replace  \JL  by      '  and   use  (  1  )  and  (2),  becomes 


The  special    linear  tangent   complex    is  then   in   general  (^=0) 

V>//(,,(  =  o. 

In  an  exceptional  manner,  however,  all  tangent  linear  complexes 
are  special  when  ^L'''  <i:  -~-  ()-  (  «'•  ) 

When  this  condition  is  satisfied  the  sphere  i/t  is  called  a  nhn/>i/nr 
*rl.:r,. 

'I  lie  conditions  to  he  satisfied  by  the  coordinates  of  a  singular 
sphere  are.  accordingly, 

V,/-—  M,      V/-,.  //;=<),      ^"'V//'      <i,  (7) 

whieh  expi'ess   respcctivelv   that   //,  satislics   the  fundamental  e<pia- 
t  ion  for  sphere  coordinates,  that  tin-  sphere  //  is  in  the  complex  (  I  ), 

and   that    it    i-  a   >iir_;-iilar  sphere. 

The  !aM  eipiation  also  expi'oses  the  tact  that  <•,//  are  the  coor- 
dinates (,f  some  sphei-e.  and  the  second  eijnation  tells  us  that  the 
sphere  /•  //  is  taiiLTfiit  to  the  sj  there  //,.  The  two  spheres  therefore 


SPHERE  COORDINATES  :v>"» 

define  a  pencil.  On  tin;  sjihriv  //  there  is,  therefore,  a  definite 
point  /',  tin-  center  <>t  the  point  sphere  of  the  pencil.  The  loni>  of 
/'  is  tin  x"  extent  ot  points  forming  the  .v///-/</<v  «/'  .s-///</?/A?/vV/Yx. 

In    order   to   (letennine    the   decree    of   the    surface   of   singularities 
we  shall  take  zt,  any  sphere  of  the  pencil  of  tangent  spheres  delined 

by  ii-  and  <•  >/ ,  so  that 

p.?,  =  (.<•,•  4- M.'/,.  (*) 

Substitution  in  (7)  e/ives  the  equations 

V  =0,      V      '''•''        :ii,      V     ''''•''      =  f), 

~~<",  4- XT  *- M'-.+  X)-  *"*  (.''<•+*•>* 

but  simple  linear  combinal  ions  ot    these  show  that   they  arc  eqiiiv- 

lllellt    to   the    three    equations 

V  :•    _-  ii        V  -  o        V  --(I.  ( ;i  j 

**'-i+\          ^('-t+xr 

y 

Cunverselv,  if  2.  is  unv  solution  of  ('.')  and  \\'e  place  u,=          —  > 

ft+\ 

it  is  clear  that  ?/(.  is  a  singular  sphere  ot  the  ijiiadratie  complex  (  1  ). 
Therefore  equations  (It)  are  satisfied  hv  all  spheres  belon^in^  to 
anv  pencil  of  tangent  spheres  detined  by  a  singular  sphere  //  and 
the  sphere  <',//,,  and,  conversely,  anv  sphere  \vliicli  satisfies  (  1' ) 
belongs  to  such  a  pencil. 

Let  us  no\v  adjoin  the  condition  that  2,  should  be  a  point  sphere  : 

nawel"v'  .,=  (>.  (10) 

Lip i at  ions  ( (.< )  and  (  I  0),  then,  deli ne  the  points  /'. 
Consider  now  anv  straight  line  /  delined  as  the  intersections  of 
two  planes  .17  and  .V.     Take 

2/"'-'  =  0  (HI 

as    the    equation    of   anv    linear    complex    which    has    M   as    a    ba>e 

sphere,  and 

N^  //r, -0  (!'_») 

as  t  he  equal  ion  ol  an  v  linear  complex  which  has  A"  as  a  liase  sphere. 

I  he  point    spheres  ot    the  complex    (11)  have  centers  on   .'/.   and 
the   point   spheres  of   the   complex    (111)   have   centers   on    .V.  so  that 
the  point   spheres  helone/inn-  to   .17  and   A'  have  centers  on  the  line  /. 

I 1  ei  ice  t  he  simultaneous  soli  it  ions  of  equal  ions  (  !'  ).   (  1 ( '  ).   (  11  i. 

aild     (\'l)     M'ive     the     point     Spheres     \\-hose     center-     lie     both     oil     the 

surtaee   of   siii<>-ii  larii  ies   and    on    ilie    line   /.     'I'hc    inimbei-   of   these 


356  FOUR-DIMENSIONAL  (JEOMETRY 

solutions  is  the  number  of  points  in  which  /  meets  the  surface  of 
singularities;  that  i>,  the  decree  of  the  surface. 

To  solve  these  equations  we  mav  begin  by  eliminating  X  from 
the  last  two  of  equations  (I').  Since  the  third  equation  of  (\\ )  is 
the  derivative  of  the  second  with  respect  to  X,  the  elimination  of 
X  skives  the  condition  that  the  second  equation  should  have  equal 
roots  in  X.  Since  the  second  equation  in  ( '.' )  is  of  the  fourth  order 
in  X,  liv  virtue  of  the  first  equation  in  (  U  ).  tin-  result  of  the  elim- 
ination of  X  is  an  equation  of  the  sixth  decree  in  .:'f  or  the  twelfth 
decree  in  zt.  This  equation,  combined  with  the  first  of  equa- 
tions (  1' )  and  the  linear  equations  (1"),  (11),  (1:1),  gives  twenty- 
tour  solutions.  Therefore  th>'  I'^mit'uni  <>f  ximjuhiritu's  tn  <>t  t/nj 
tit'i'nti/-f"urtJi  "/•</»•/•. 

Equations  ((.))~(1-)  niay  be  otherwise  interprete<l  by  consider- 
ing (11)  ami  (  1 -! )  as  the  equations  of  two  complexes  with  base 
spheres  which  are  not  planar  and  therefore  intersect  in  a  circle. 
which  may  be  any  circle.  The  special  spheres  of  the  complexes 
have  their  centers  on  this  circle,  and  the  special  spheres  which  also 
satisfy  (7  )-('.*)  are  point  spheres,  since  the  condition  that  they  be 
planar  adds  a  new  equation  which  in  general  cannot  be  satisfied. 
Hence,  by  the  argument  above,  any  circle,  as  well  as  any  straight 
line,  meets  the  surface  of  singularities  in  twenty-four  finite  points. 

If  the  equations  arc  expressed  in  ('artesian  coordinates,  the 
circle  will  meet  a  surface  of  the  twenty-fourth  order  in  forty-eight 
points.  We  have  accounted  for  twenty-four  finite  points  :  the  other 
twenty-four  must  lie  on  the  imaginary  circle  at  infinity.  Since  the 
plane  of  the  finite  circle  meets  the  circle  at  infinity  in  two  points. 

We  have  the  theorem:  Tin-  *///_-/i/rv  ,if'  x/tn/K/itntirx  i-^/tfilt/t*  //!>' 
i HKi'jinitrif  '•//•'•/»'  lit  iittinttij  <(a  i(  lu'i-li'i'fnlil  /nn'. 

Return,  now.  to  the  pencil  (  >>  ).  There  is  one  plane  /<  in  the 
pencil  which  is  tangent  to  //  at  /'and  is  uniquely  determined  l>v 
//  .  Such  planes  form  an  s,~  extent  which  envelop  a  surface.  To 
rface  is  the  surface  of  singularities  let  //,+  <///,  be 
neighboring  to  // .  so  that 

V/^/v^u,      V -•,/,<///,    -  o.      ^ ••;//.'///,    -  ".  (1;>>) 

'I'he  pencil  of  tangent  spheres  defined  by  //  -f-  •///  and  <\(  i^  4-  *1  /i  )  \< 

i'         (  <•    *    n  )  (  //    ^    '///    ),  (It) 


SPHKKK  COOKWXATES  :\:,~ 

and  (In-  condition  that  i\  should  he  tangent  to  .:,  is  satisticd  1>\ 
virtue  of  (  7 )  and  (  1  •> ).  Hence,  in  piirticular,  the  point  /',  the  center 
of  the  point  sphere  of  (  S  ),  lies  in  the  plant-  /<'  of  the  pencil  (14); 
that  is,  /'  is  the  limit  point  of  intersection  of  t\vo  neighboring 
planes  j>  and  is  thereloiv  a  point  of  the  surface  enveloped  bv  j>. 
This  establishes  the  identity  of  the  surface  which  is  the  locus  of 
/'  and  that  enveloped  by  />. 

The  class  of  the  surface  ot  singularities  is  the  number  of  the 
planes  ]>  which  pass  through  an  arbitrary  line.  To  determine  this 
number  we  may  again  set  up  equations  ( (.i ),  (11),  and  (1-),  but 
replace  (10)  by  Ml+/w6=0,  (lo) 

which  is  the  condition  that  nt  should  be  a  plane. 

Anv  plane  of  cither  of  the  complexes  (11  )  or  (  1  -  )  intersects 
the  base  plane  M  or  A' respectively  in  a  straight  line,  and  therefore 
the  planes  common  to  .)/  and  A"  pass  through  the  line  /.  The  solu- 
tions of  equations  (I1).  (  1  1  ),  (  1  -  )•  and  (  1  ~> )  give,  therefore,  the 
planes  tangent  to  the  surface  of  singularities  which  pass  through  /. 
Hence  tin'  mtrfiti'?-  nf  Kinijuhiritu'n  i*  of  the  tirentij-fuurth  i-l'ixx. 

154.  Duality  of  line  and  sphere  geometry.  Since  line  coordinates 
and  higher  sphere  coordinates  each  consist  of  the  ratios  of  six  quan- 
tities connected  bv  a  quadratic  relation,  there  is  duality  between 
them.  To  bring  out  the  dnalistic  properties  we  shall  interpret  the 
ratios  of  six  quantities  ,/v  connected  bv  the  relation 

->V  +  ./•;  +  .'•:  +  .'••  +-  .'^  +  .r,:  =  «.). 

on  the  one  hand,  as  the  sphere  coordinates  <i t  of  jj  147  and,  on  the 
other  hand,  as  the  Klein  line  coordinates  of  >j  loH. 

It  is  to  be  noticed  that  for  a  real  line,  as  shown  in  vj  l^O.  \\'e 
have  j-.  .i':.  jr  real  and  ./  .,  ./•. .  .rf  pure  imaginary.  (  )n  the  other 
hand  it  follows  from  vj  Jj  1  4t>,  147  that  for  a  real  sphere  \\  e  have 
./•,.  ./•.,,  ./•  .  ./•  real  and  ./'.,  ./ ,  pure  imaginary.  Hence  configurations 
which  are  real  in  either  the  line  or  the  sphere  space  \\ill  lie 
imaginary  in  the  ollr.-r. 

It  is  also  to  be  noticed  that  a  sphere  for  which  ./\=0  is  peculiar, 
being  a  special  sphere,  but  the  line  tor  which  j'f  ()  has  no  special 
geometric  jii'opcrt  ics.  The  complex  ot  lines  r  (i  has.  however, 
a  peculiar  ride  m  the  duah>tic  relations.  \\  c  shall  call  this  com- 
plex ( '.  Its  equation  m  IMiicker  cnordinates  i<  i>  />  =  ". 


:;:>s  FOLK  DIMENSIONAL  GEOMETRY 

Two  spheres  whose  coordinates  ditler  only  in  the  sign  of  ./•  are 
the  same  in  the  elementary  sense,  hut  two  lines  whose  coordinates 
differ  in  the  same  way  are  distinct  and  conjugate  with  respect  to 
the  complex  ('.  The  relation  between  sphere  and  line  is  therefore 
in  one  sense  one-to-two,  but  becomes  one-to-one  by  the  convention 
of  distinguishing  between  two  spheres  which  differ  in  the  sign  of 
the  radius. 

Any  sphere  for  which  ./^  +  /./',—  0  is  a  plane,  but  the  correspond- 
ing line  has  no  special  geometric  property.  The  complex  of  lines 
./•  +  i.r.  il,  however,  will  ha\e  a  peculiar  role  in  the  duality.  We 
shall  call  this  complex  .V.  It  is  special  and  consists  of  lines  inter- 
secting the  line  with  coordinates  1:0:0:0:  /.  Its  equation  in 
IMiicker  coordinates  is  //.,(  o. 

We  have  now  as  immediate  consequences  of  our  previous  results 
the  following  dualistic  relations: 

/*'•/•'  ^'       / 

A  st  raight  line.  A  sphere. 

A   line  of  the  complex  ('.  A   special  sphere. 

A    line   of  t  he   complex   X.  A    plane. 

A  line  of  ('  but   not  of  X.  A  point  sphere. 

A  line  of  s   hut   not  of  ('.  An  ordinary  plane. 

A  line  of  ('  and  of  X.  A  minimum  plane. 

Two  lilies  conjugate  with  respect  Two  spheres  (littering  only  in 

to''.  the  sign  of  t  he  radius. 

Two  intersecting  lines.  Two  tangent   spheres. 

A  noiispecial  complex.  A  nonspecial  complex. 

A  ^pecial  complex  consisting  of  A  special  complex  consisting  of 

lilies  intersect  ing  a  fixed  line.  spheres  tangent  to  a  tixed  sphere. 

A  linear  congruence  consisting  A  linear  congruence  consisting 

of  lines  intersecting  two  lines.  of  spheres  tangent  to  two  spheres. 

A  linear  series  lorining  one  set  A  linear  series  forming  one  of 

ni  generators  of  a  qiiadrie  surface.  the  families  of  spheres  which  eii- 

\  eh  1)1  a  1  hipin's  c\  elide. 

A  quadratic  line  complex  with  A  quadrat  ic  sphereeoniplex  with 

it>  Mir_;ular  surface.  its  singular  .surface. 

A  pencil  ot  lines  corresponds  to  a  pencil  ot  tangent  spheres, 
and  ,i  bundle  of  lines  to  a  bundle  of  tangent  spheres,  ('onsider  a 
point  /'  and  the  /_~  lines  through  it.  'I  hev  correspond  in  general 


SPHERE  COORDINATES 

to  a  bundle  of  tangent  spheres  which  have  in  common  a 
line  i>  Ck.  !-~>L!).  It  is  therefore  possible  in  this  way  to  set  up  a 
correspondence  of  the  line  space  and  the  sphere  space  by  which 
anv  point  of  the  line  space  corresponds  to  a  minimum  line  of  the 
sphere  space. 

An  exception  occurs  when  the  point  /'  of  the  line  space  lies  on 
the  axis  of  the  complex  \.  Then  all  lines  through  /'  belong  to  \, 
and  the  corresponding  bundle  of  spheres  consists  of  planes  which 
have  in  common  only  a  point  on  the  imaginary  circle  at  infinity. 

('oiisider  two  points  /'  and  (t>  connected  by  a  line  /  correspond- 
ing to  a  sphere  x.  /'corresponds  in  the  first  place  to  a  bundle  of 
spheres  containing  *  and  therefore,  in  the  second  place,  to  a  mini- 
mum line  j>  on  x.  Similarly,  <t>  corresponds  to  a  minimum  line  </• 
also  on  >•.  If  ji  and  •/  intersect  in  a  finite  point  .)/.  the  point  sphere 
with  center  .17  belongs  to  both  the  bundle  of  spheres  containing/' 
and  that  containing  >/.  Therefore  the  line  corresponding  to  this 
point  sphere  must  pass  through  /' and  (J.  Hence  /,  since  it  corre- 
sponds to  a  point  sphere,  is  in  this  case  a  line  of  the  complex  < '. 

('onversclv.  it'  /  is  any  line  of  the  complex  ('  the  minimum  lines 
corresponding  to  /'  and  (>  lie  on  a  special  sphere  and  intersect. 

<  hherwise,  if  /  is  not  a  line  of  the  complex  ('  the  minimum  lines 
do  not  intersect  in  a  finite  point  and  hence  are  two  generators  of 
the  same  family  on  x. 

('oiisider  now  the  line  /'  conjugate  to  /  with  respect  to  the  com- 
plex ('.  '1  he  points  of  this  line  correspond  to  generators  of  the 
same  sphere  x.  Hut  points  of  /  and  /'  arc  connected  by  a  line  of  <\ 
and  therefore  the  '4'eiicrat  ors  ^iveii  by  /'  intersect  those  C.-IYCH  by  /. 
Therefore  the  generators  '_;-iven  by  points  of  /  and  /'  belong  to 
different  families. 

('oiisider  now  the  lines  of  a  plane.  They  form  a  bundle  which 
corresponds  to  a  bundle  of  tangent  sphere--.  It  is  therefore  possible 
to  set  up  a  correspondence  of  line  space  and  sphere  space  hv 
which  a  plane  corresponds  to  a  minimum  line.  \\  e  have  nothing 
new,  however,  since  the  lines  which  lie  in  a  plane  are  conjugate 
with  respect  to  ('  of  the  lines  which  pass  through  a  point.  In  fact, 
it  we  keep  tii  the  correspondence  (if  point  and  minimum  line  it  is 
not  difficult  to  show  that  the  f  points  of  a  plane  correspond  to 
/.  "  m  mini  u  in  lines,  \\  hie  h  can  be  arranged  in  /_'  spheres  which  base 


300  FOfK-niMKXSlONAL  (JKOMETKY 

a   niiiiiiuuin   line   in   common,   so   that   in   this   way   a  plane  corre- 
sponds to  a  minimum  line. 

\Ve  may  exhibit   these  results  in  the  following  table: 


A  point.  A  minimum  line. 

The  points  of  a  general  line  /.  One     set     of     generators     of    a 

sphere  N. 

The  points  of  /'  conjugate  to  /  The  other  set  of  generators  of  ,s. 
with  respect  to  < 

The   points  on   a   line  of  f  but  The   minimum  lines  on  a  point 

not  of  X.  sphere    (the    lines    of  a    minimum 

cone). 

The  points  of  a  line   of   X   but  The  two    families   of   minimum 

not    of    ('   and    the    points   of  the  lines  of  a  plane. 
conjugate   line   with   respect   to  ('. 

The  points  of  a  line  common  to  The  single  family  of  minimum 

('  and  X.  lilies  on  a  minimum  plane. 

Consider  now  any  surface  /*'  in  the  line  space.  We  may  find  a 
corresponding  surface  in  the  sphere  space  as  follows.  Let  /'  be 
any  point  on  /-'and  consider  the  pencil  of  tangent  lines  to  /-'at  /'. 
These  lines  if  infinitesimal  in  length  determine  a  surface  element. 

Corresponding  to  the  pencil  of  tangent  lines  there  is  in  the 
sphere  spare  a  pencil  of  tangent  spheres  which  determine  a  point 
/''  and  a  tangent  plane:  that  is,  another  surface  element.  It  may 
be  noticed  that  the  point  /''  is  the  center  of  the  point  sphere  which 
corresponds  to  the  line  of  the  complex  ('  in  the  pencil  of  lines 
which  lie  in  the  surface  element  of  /•'. 

We  haye  in  this  way  associated  to  a  surface  element  in  the  line 
space  a  surface  element  of  the  sphere  space.  When  the  surface 
elements  in  the  line  space  are  associated  into  a  surface  /•'.  the  sur- 
face elements  in  the  sphere  space  form  another  surface,  /•'',  which 
corresponds  to  /•'. 

To  any  tangent  line  of  /-'  at  /'corresponds  a  tangent  sphere  of 
/•''  at  /''.  It  is  known  from  surface  theory  that  coiiseciitiye  to  /' 
tin-re  are  two  points  <t)  and  //  mi  /•'  such  that  a  tangent  line  at 
either  coincides  with  a  tangent  line  at  /'.  The  tangents  J'tt>  and 
/'//  are  the  principal  tangents  at  /'.  It  the  directions  ot  one  of 


SI'HKKK   COORDINATES  3d 

these  tangents  is  followed  on  tin-  surface,  we  have  a  principal 
tangent  line  (or  an  asymptotic  line)  on  /•'. 

Corresponding  to  this,  there  are  in  the  sphere  space  two  con- 
secutive points  (/  and  /.''  on  /•''  such  that  a  tangent  sphere  at 
either  coincides  with  a  tangent  sphere  at  /''.  If  one  of  the  direc- 
tions /''(/  or  /''//'  is  followed  on  /•'',  we  have  a  line  of  curvature 
of  /•". 

'1  hcretore,  in  ttte  <'i>rr<'xj>'>/t<t<'n<'t'  bt'for?  itx  principal  tii/ti/eitt  lutes 
"/i  it  xurt'ii'-f  in  tin'  lint'  xjxti'e  1'vrrt'sj.xttnl  t«  lim-x  <//'  curcature  uii  the 
cui'i't'spuiidinij  xtt/'tcice  in  the  xp/itjf'tj  xfnti't'. 

EXERCISES 

1.  Show  that  the  relation   between  line  space  and  sphere  space  mav 
be  expressed   bv  the  equations 

-  /.::  =  '/'.,•  -  (X  -  il')t, 
(X  +  !)•);:  =  V//  -  Z/, 

where  .r  ://:-;:  t  are  ('artesian  point  coordinates  in  the  line  space  and 
A  :  }'  :  '/.  :  7'  arc  similar  coordinates  in  the  sphere  space.  Verity  all  the 
results  of  the  text. 

2.  Trace   the   analogies    between    the    four-dimensional   sphere  geoiu- 
etrv    and    the    three-dimensional    point    geometry    with    peiitaspherical 
coordinates. 


rilAlTKR   XIX 

FOUR-DIMENSIONAL  POINT  COORDINATES 

155.  Definitions.  We  sluill  now  develop  the  elements  of  a  fonr- 
dimeiisional  geometry  in  which  tin1  ideas  and  iiH'thods  of  the  ele- 
nu'iitarv  three-dimensional  point  geometry  are  nst-d  and  which 
stands  in  essentially  the  same  relation  to  that  geometry  as  that 
docs  to  tin-  <4vometrv  of  the  plane. 

We  shall  define  as  a  [mint  in  a  four-dimensional  space  any  set 
of  values  of  the  four  ratios  .1^:  .r,  :./',:  ./4:  ./-.  of  five  variables.  In  a 
nonliomogeneous  form  the  point  is  a  set  of  values  of  the  four 
variables  (  ./•,  //.  z,  ic  ). 

A  straight  linr  is  defined  as  a  one-dimensional  extent  determined 
hv  the  equations 

pjv=//.+  X,r1,          (/  =  !,  -2,  :',,  4.  f>)  (1  ) 

where  //  and  z:  are  two  fixed  points  and  X  is  an  independent  variable. 
A  [dan*'  is  defined  as  a  two-dimensional  extent  determined  l>v  the 

equations  •      1    .>    .»    <    -  ,  ,  .>  , 

p.i\  —  //,  -f  X^,  -f  nti\,  (<  =  1,  -,  3,  4,  ;>  )  (  '1  ) 

where  j\,  //(,  .?,  are  three  fixed  points  not  on  the  same  straight  line 
and   \.  ft  are   independent  variables. 

A  liiffH-i'^litHf'  is  defined  as  a  three-dimensional  extent  determined 
by  the  equations 

p.i-  -=  //f  -f  X^-,  +  n.n\  -f  I'HS  (  i  —  1,  -,  :>>.  4,  ")  )  (  :>  ) 

where  //,,  ^(,  n\,  a,  ai'e  four  fixed   points  not    in  the  same  plane  and 
X.  /z.  r  are  independent  variables. 

From  these  definitions  follows  at   mice  the  theorem: 


/.    .1    xtt'iti'i/Jtt    II  in'    ix    rntnph'trlif    <tn<l    Uni<jUi'lif    •/•  (<rnn  n<<l  //_//   <i/it/ 

fir,,    i,f'   //.-•    jK-i/ttx,    <i    fililin     lul    itmj    tli/'tf    "f   it*    fmiiittt    ll'hit'li  <!/'<'    //"/ 

i-nllnn  it  r.   iiml   ii   //  i/jn  rjilnin'   In/  inn/  J"iir  »j'  it*  jio/iit*  trhti'h  ii/'f   n»f 
t-n^liinnr. 

The  forms  of  e<  i  nations  (  1  ),  (  l'  ),  (  :i  )  show  t  hat  if  the  fixed  points 
are  Li'iven,  the  eorresjidiiding   loetis   is  completely  determined.     The 


IMUNT  COORDINATES  :;j;;j 

theorem  asserts  that  any  points  mi  the  Incus  which  arc  the  >ame 
in  number  and  satisfy  the  same  condition  as  the  ^ivcii  points  mav 
be  used  to  detine  the  locus.  \Ve  shall  show  this  for  the  plane  (-). 
Let  K  be  a  point  deiined  by  e<{iuitions  ( '2  )  when  \  —  \^  fi  —  fj.  ; 

that    is.   let 

},=  //,+  \z,+  /*,«',.  (  1  ) 

K<[iuttions  (-2)  may  then  be  written 


which  are  of  the  type     p->\—  ) ',  +  ^-X  +  P  "\-  ( •"> ) 

Then  anv  point  wliich  can  be  obtained  from  (  ml )  can  also  be 
obtained  from  (•">),  and  converselv. 

The  discussion,  however,  assumes  that  T  is  not  on  the  same 
straight  line  through  zt  and  n\;  for  if  it  were,  the  coordinates  of  )" 
would  not  be  of  the  form  (4).  In  fact,  to  obtain  from  ('2)  points 
on  the  line  //,:,  in  the  plane  ( '_' )  it  is  necessary  to  replace  X  and  /z 

bv  the  fractional  forms  -,      .  write  the  equation   of  the  plane  as 

v     v 


and  then  place  v  =  0. 

\\'e  liave  shown  that  in  equations  (  '2  )  the  jtoint  //,  may  be 
replaced  hv  an\-  point  not  nn  the  same  straight  line  \\'ith  zt  and  >r  . 
In  the  same  manner  each  of  the  other  points  mav  be  replaced,  and 
the  theorem  is  proved  for  the  plane. 

The  student  will  have  no  dit'liciiltv  in  proving  the  theorem  for 
straight  line  and  hyperplane. 

Another  immediate  consequence  of  the  definitions  is  the  theorem  : 

II.  If  f ii'n  jiniiilK  In'  in  it  ji]<ni>\  tin'  /fni'  <l>  t'nniii' <l  f'i/  tli.  in  //.  .v 
///  ///••  jiliim •  ;  If  fht't'i'  /n'iiifx  In'  in  it  fi i/jufjil'i i>',  tin  ji/<//n'  if,  f,  /•/////"'/ 
I'll  tin  nl  III-K  III  thi'  liinufnlilili. 


RU'K    m.MKNSIONAL  (JKOMKTKV 


1  lence  : 

///.  A ni/   hi/per i>1  ane    mai/  he    represented   hi/   a   linear   equation    in 
the  coordinates  j\. 

( 'onyersely  : 

IV.  An >/  linear  equation   in  .r,  represent*  a  hi/perplane. 

Let  2'V,=  n  (7) 

be  such  an  equation,  and  let  //,,  zt,  ir.,  nt  be  four  points  satisfying 
the  equation  but  not  on  the  same  straight  line.    Then  we  have 

2llt,  —  0     y,,  ?  =  o     Y,rN'  =  o     V,M/ =  o 
i,/.-  ^  r.  £/  '"  <  Zj  *  ' 

and   by  eliminating  >r  from  these  (Mjuations  and   (7)  we   have   an 
equation  of  the  form  (fi)  and  thence  equations  of  form  (•>). 

If   we   eliminate   p,  \,  ft  from  equations   ('!)  we   have  the  two 
equations 

•'-,     //,      *,      "V 


I          -'4  4  4  :.          •':,  5 

That   is: 

V.  An >/  plane   ntai/  he  represented  hi/  tim   //near  equations  in   the 
coordinates  ,'\. 

( 'onverselv  : 

VI.  An//  tii'n  indepemlent  linear  f  quaff  nun  represent  a  plane. 

be  such  equations.    Since  they  are  independent,  at   least  one  of  the 
determinants       '       '     is  not   /ero.     Let    us  assume   that       *       :'    —  0. 

1  '"  4 

The  two  eijiiations  can  then  be  solved  lor  ./'  and  ./•.,  and  thus 
reduced  to  two  of  the  type  ( ',»  )  with  tf.— 0  and  /»  =  0.  I  f  //,..:,,//', 
are  three  points  satisfying  the  equations  but  not  on  the  same 
straight  line,  we  mav  then  eliminate  <<t  and  <\  and  obtain  e<jiiations 
of  the  form  (S)  and  linallv  of  the  form  ('_')• 
In  the  same  manner  wt-  may  easilv  jn'o\e: 

VII.  A  ni/  xtraia/tt  ///<<•  mai/  /-<•  re/irem'nted  /-//  three  linear  equating, 
and  an//  tJiree  independent  linear  i'qii< items  ri'/i/'i'senf  <i  xtraiaht  hue. 


POINT  COORDINATES  .'Jf,:, 

As  a  special  case  of  theorem  IV,  any  one  ol  the  live  equations 
.r  =  0  represents  a  hyperplane.  ('(insider  in  particular  r=  0.  The 
points  in  this  livperplane  have  the  coordinates  ./•:./•,:./•;./•,  as  in 
project  ive  t  liree-diniciisii  mal  space,  and  the  definitions  of  straight 
line  and  plane  are  the  usual  definitions.  The  two  equations  which 
represent  a  plane  consist  ot  the  equation  ./•.  =  "  and  anv  other  linear 
equation.  If,  then,  the  equation  j~.  =  0  is  assumed  mice  for  all,  a 
plane  is  represented  by  a  single  equation.  Similarly,  a  straight  line 
in  ./•.  =  (i  is  re  ]i  resell  ted  by  t  wo  equat  ions  besides  the  CM  ]  nation  x,  =  <>. 
Obviously  the  difference  between  the  representations  of  a  plane  in 
three-dimensional  and  four-dimensional  geometrv  is  similar  to  that 
between  the  representations  of  a  straight  line  in  two-dimensional 
and  three-dimensional  geometry. 

.lust  as  plane  geometry  is  a  section  ol  space  geometrv,  so  space 
^eoniet  rv  is  a  section  of  four-dimensional  geometry,  the  three- 
dimensional  space  being  a  hvperplane  of  the  four-dimensional  space. 

156.  Intersections.  \\  e  shall  proceed  to  give  theorems  concerning 
the  intersections  of  lines,  planes,  and  hvperplanes.  In  reading  these 
it  mav  be  helpful  for  the  student  to  bear  in  mind  that  within  the 
same  livperplane  these  theorems  are  the  same  as  those  of  the  ordi- 
nary three-dimensional  geometry.  but  differences  emerge  as  we 
consider  figures  in  different  hvperplanes. 

/.  Tn'n  //  v/"  /•/'/''//<•*  inti'W'f  in  ii  /'/<i/i<'.    All  liypi't'pJinii'n  tJiroui/h 

f/n-  mini!'  j'liini1  f'li-in  n  fn'/ii'if,  iiit'1  <ni  if  ?"'"  i  if  tlu'Sf  //  ///',  r/1/iint's  ///»/>/ 

/„>     IIS.'J    tn     ,1,'fuU'     t)l>'    }l!<ttl<>. 

The  first  part  of  this  theorem  follows  immediately  from 
theorem  VI,  £  1  .V>.  For  the  latter  part  we  notice  that  anv  hvper- 
planes of  the  pencil  V\/  ./•  4-  \"V  //  .>•---(}  intersect  in  the  plane 

^^*      '  J^W 

determined    bv    V'/,r-=  0    and    V/,i./-i=0. 

II.  Thr'i  /ii/iifffitiini'K  tint  in  tin-  mi  ii>i'  jH-Hi'tl  uiffWi't  in  <i  xfriiii/hf 
////>'.  A//  /ii/jti'i'ft/tnn'N  flir<»i<//t  f/i>'  mt  mi'  I/DI'  fun/I  n  1nniiJlt\  ii/nl  <DH/ 
tlir'i  "t  tl/i'i/i  nut  in  f/n'  xii/t>i'  i>i'iii'il  il  >t  i  run  HI'  tin1  I/HI', 

This  follo\\-s  at  once  from  theorem  VII,  ^  1  ">">.  The  bundle  of 
hverlanes  is  i\cn  bv  the  euation  ^<ij-\-  X^^.r  -f-  ^"^  ••./  ". 


///.   /•'-///•  fn/i>t')'nl<i>n'x  not  in  tin1  xitnit'  linii'lli    i  /it  *  i'.*'  i-t   /i/   ii  /'"/lit. 
.  1  1  !  !i  i/  /'<  rjiJ'i  ni  N  t/i/'nin/fi  t  In'  fit  an'  i  mi  nt  >  at-  Hi  1  1  t  Ii  TI  i'-il  i  a"  a  *i  "ii  <il  *  .rti'  nt, 

/lll'l     illll/      /"///'     nt'    tlll'lll      lint      III     till       Silllli'     /ill/I'//'1     llt'tl'/'l/llltt'     tin          '"'/!/. 


1-ori;    DIMENSIONAL  <  i  K(  >M  KTR  Y 

This  follows  from  tin-  tact  that   the  tour  equations 

ViV,=  0,         V/,  ,/•  =  <>,        \V./-      .11.         V,/.;.=<> 

determine  in  general  a  single  point.     The  exceptions  are  when  the 
four  equations  represent   hvperplanes  ot   the  same  bundle. 

IV.  A  j'l'i  ni'  <in</  it  hi/jit'i-jilii  in'  iiltf'l'st'i't   i it  <(  xtt'(ri<//tt  liin    ii nil  xx  tin' 
plitii'  lit  .s1  i'/itiri'/i/  in  (In'  hi/^t'r/tlum: 

For  the  equations  which  determine  the  points  common  to  a  plane 
and  a  hvperplane  are  three  linear  equations  which  in  general  deter- 
mine a  line.  If,  however,  the  plane  lies  in  the  hvperplane,  the  lat- 
ter niav  lie  taken  as  one  of  the  equations  of  the  plane  (theorem  I  ), 
and  we  have  only  two  equations.  Furthermore,  if  the  plane  inter- 
sect the  hvperplane  in  three  points  not  in  the  same  straight  line, 
it  lies  entirely  in  the  hyperplane  l>v  theorem  II.  v<  I-).). 

V.  T'/'"  j'liiiii'x  intt'rxi'ft   in  '/  xiiii/lt'  fi'iinf   mtfi'xx  ///»-//  //«    in  tin-  minif 
h '/)"  rjil'tni'.     In  t/i<if  i-iixf  tin •//  //it*  rxiff  ///  ,/  //'//,-,  <,,•  ,•.//'/,,•/<//. 

For  the  points  common  to  two  planes  muM  in  general  satisfy 
four  linear  equations  and  hence  reduce  to  a  single  point.  It.  how- 
ever, the  planes  are  in  the  same  hyperplane.  the  equation  of  that 
hyperplane  may  be  taken  as  one  ot  the  equations  of  each  of  the 
planes,  and  the  points  common  to  them  have  only  to  satisfy  three 
equations.  Furthermore,  if  the  two  planes  intersect  in  a  line,  the 
hvperplane  determined  by  four  points,  two  on  the  line  of  inter- 
section and  one  on  each  ot  the  planes,  will  contain  both  planes 
(  theorem  1 1,  ^  1-V>  ). 

VI.  Tl/'i •>•  filinii-x  H"t  in  f/h'  xiit»i'  ]iii)>iT]iliijit'  if"  n"f  in  //«  a-  /''i/  i/iti  /•- 
,v.  ft,  fiiif   in<ii/  i i/tiTxi'ft   in  ii  xni;il>'  ji'ii/it  "/•  in    n   xtriiiijlit  fi/if.     Tln't'i' 
jilitihx  in   tin-  xi in) i'  /it//>i'/'/i/<i)it'  //ifiTxi •!'(  in  ii  j>«int  «r   <>  xtf<i!;/iit  liih'. 

The  points  of  intersection  of  three  planes  must  satisfy  six  equa- 
tions, which  is  in  general  impossible.  If  the  planes  are  in  the  >ann' 
hvperplane,  however,  the  number  of  equations  is  reduced  to  \\\  least 
four  by  taking  the  equation  of  that  hyperplane  as  one  ot  the 
equations  of  each  of  the  three  [danes. 

lint  consider  four  hvperplanes  intersecting  in  a  point.  It  is 
possible  in  a  number  of  wavs  to  pair  these  hvperplanes  so  as  to 
determine  three  planes  which  have  the  point  in  common  but  are 
imt  in  the  >amf  hvi>en>lane. 


POINT  rouiiniXATKs  :;r,7 

Or.  apiin,  consider  anv  two  jilaiics  intersect  inij  in  a  point  A. 
It  is  easily  possible  to  select  t\vo  points  /.'and  ('  which  shall  not 
lie  in  the  same  hvperplane  with  either  of  the  LMVCII  planes.  'I'he 
plane  A  l'>i'  has  the  point  .1  in  common  with  the  first  two  planes, 
l»nt  they  do  not  lie  in  the  same  hyperplanc. 

Similarly,  let  two  planes  intersect  in  a  line  A  ft.  A  plane  may  In- 
passed  through  .///and  a  point  ('  not  in  the  same  hyperplane  with 
the  first  two  planes.  ()!'  course  any  two  of  these  planes  lie  in  the 
same  hypcrplane  (theorem  V  ). 

VII.  .1  xtrii'ujlit  1  1  'in1  inn/  ii  lii/i'i'i-jil'iin'  inti'i-xi't't  in  <>  xin</ff  jmiitt 
ini/i'xx  tin'  lini1  l/'i'x  t'litit'ffi/  iii  tin1  /ii/HTifunc. 


e   reason     s  oyoiis. 


VIII.  .  I  atriiiijht  //in'  ii>i</  <i  i>l<nn'  ilu  ii"f  iiiti'rxt'i-t  ini/i'xx  tin-it  //»• 
///  (In'  xiinii'  hi/fii'i'nftini'.  In  tin'  /ii//i'/-  i-iixf  thiii  I'iftn'r  fnft'rxi-i-f  in  it 
>u//tf  in'  (hi1  //iii'  //i'x  i  ntii'i'li/  in  tin'  il'i/ii1. 


The  points  common  to  a  straight  line  and  a  plane  must  satisfy 
fiye  equations,  which  is  in  evneral  impossible.  If.  howeyer.  the  line 
and  plane  are  in  the  same  hyperplane,  the  number  of  equations 
may  be  reduced  to  tour. 

Attain,  let  the  line  and  plane  intersect  in  the  point  .1.  Three 
other  points  may  be  taken  :  /•'  on  the  line,  and  < '.  />  on  the  plane. 
The  h\  perplane  determined  by  .1,  />'.  <\  l>  then  contains  both  the 
line  and  the  plane. 

IX.     Tiro  *t /•</!>  / /it  lilli'X   tin  lint   nttl'l'Xi'i-t    Htlfi  XX  till    I/  I'll'   ill   till'  Xil  I  111'  f  'III » i '. 

In  tin'  liitti-r  i-ftxi'  f/n'if  inti 'fx,'i-f  in  <i  /"/////  «/•  r»lin'nli    / />r«i/i//tnnt. 


I  he  points  common  to  two  hues  must  satisfy  six  equations, 
which  is  in  general  impossihle.  If.  howeyer,  they  lie  in  the  same 
plane,  the  number  of  equations  may  be  reduced  to  four. 

A'_;'aiii,  let  the  two  lines  intersect  in  a  point  .1.  The  plane  deter- 
mined by  .1  and  two  other  points,  one  on  each  hue.  contains  both 
lines. 

\\  e  close  tins  section  with  two  theorems  on  the  determination 
ot  planes  and  hypcrphuies  which  ha\c  already  been  foreshadowed. 

X.   A  /i/'t/ii    ntti it  /«•  ili'ti'firt'nii'tl  l^i  (  1  )  t/i/'i'i1   ii"t/itx   not   ni  tin1  xit//ii 

Iliii'  ;     (   -  )    'I    1 1  in-    illhl    il    [mint    il"t    nil    it  ,'     (   •')  )    /"'"    ////<  /'Xi  i'tlll'1    lllli'X. 


JitiS  F»»l'i;    DIMKNSIONAI,  (JKOMKTKY 

XI.  A  In/I"  i'i>l<nif  niit/i  I"'  ili-tcrinini'il  f>//  (  1  )_/«///•  points  »<>f  in  tin* 
tut/Hi  I:/<IH.  :  (  -  }  'i  i'/'ii/>  '//i'/  it  i>»int  imt  nil  It  :  (  '.\  >  ,1  />l<ni<'  nn<l  » 
//'/it-  i//fi  •/•*>  <•?/'//</  i<  :  (  1)  t/i'n  II/,I//IK  hiffWrfiHi/  in  ii  ///if;  (  .">  )  tJifft' 

llllix    ll"t    in    tin1    inliiif    jililllf    intl'THt'cttHt/    I/I    <l    fi'iint. 

157.  Euclidean  space  of  four  dimensions.  We  shall  consider  now 
a  t'onr-diniriisional  space  in  \\liich  metrical  propcriifs  analogous  to 
those  of  tliivr-dinifiisioiial  Kiiclidcan  space  are  assumed.  For  that 
purpose  let  us  replace  the  ratios  .r  :  .ra:  j- :  .r  ;  j\  bv  ./'://:  z  :  /r  :  f. 


Then  if  f  -'- 'I  the  coordinates  ,V.  )'.  /,  If  are  finite,  and  the  values 
(  .V.  )".  /.  //'  )  are  said  to  represent  a  point  in  tinite  space.  If  /  =  0 
one  or  more  of  the  coordinates  A'.  )".  /..  II"  is  infinite,  and  the  ratios 
.r://:^:/r:0  are  said  to  represent  a  point  at  intinitv. 

•  lit 

Tin-  1'iiinitt'tin  f     -'•  r>  jit'' xfiit*,  tlnii,  ///-'  liifjH'i'jtliitn'  'it  /'nti/i/t //. 

'I'hc   i//'tiftttiff  lietu'eeii    t  \\  o   points   is  detincd   1»\'   the  ('((nation   in 
the  nonhttnio^eiieous  coi'irdmates 


from  which  it  appears  that  the  distance  between  two  tinite  points 
i--  tinite  and  that  the  distance  between  a  finite  point  and  an  infinite 
point  is  in  general  infinite. 

The   equations   ot    a   straight    line   are   in    noiihomoj^eiieous   coor- 
dinates 

1    •    \  1  +  X  1   f  X 

,., 

.V,      A',       }'..      >',       /...      /,       II '.      I/, 

hieh    ma  v  1  ic   wrii  len 

\    \.     }     r,     /.    /.,     ii'    n, 
//          f  /> 


POINT  COORDINATES  :;ij'.i 

line.  It  is  mulilv  seen  that  a  line  inav  be  drawn  through  the 
point  (A",  }' ,  Z,  IT  )  with  any  given  direction  and  that  two  lines 
through  that  point  with  the  same  direction  coincide  throughout. 
There  is,  therefore,  a  one-to-one  relation  hetween  the  lines  drawn 
through  a  iixed  point  and  the  ratios  we  have  used  to  define  direction. 
This  justifies  the  use  of  the  word. 

Two  lines  with  the  directions  .1  :/•':'":/>  and  .  /  o:  //,:  f\t:  />, 
respectively  are  said  to  make  with  each  other  the  angle  #,  defined 
by  the  equation 

A,  ,/.,+  /;,/,•.,+  (\('.,+  />,/>., 
cos  V  = —  •  (h  ) 

VA?+  ii'-  +  r,j  -f  />,-  v  A:  +  /;.;  +  r;  +  />; 

Consider  the  hvperplane,  H'=o.  Anv  point  in  that  hvperplane 
is  tixed  hv  the  coordinates  (A".  )',  7,  ).  and  the  distance  between 
two  points  reduces  to  the  Euclidean  distance.  The  equation  of 
any  straight  line  in  that  hvperplane  is 


so  that  />  —  (}.  Hence  the  definitions  of  distance  and  angle  become 
those  of  Euclidean  distance  and  angle.  Therefore  the  geometry  in 
the  hvperplane  //'=<>  is  Euclidean. 

Similarly,  the  geometry  in  each  of  the  hvperplanes  A"=  0,  }'=<), 
Z  —  (I  is  Euclidean.  The  same  will  be  shown  later  to  be  true  for  any 
hvperplane  except  the  hyperplaiie  at  infinity  and  certain  exceptional 
imaginary  hyperplanes.  We  accordingly  call  this  four-dimensional 
geometry  Euclidean. 

In  the  hvperplane  at  infinity,  t  —  0.  a  point  is  fixed  by  the 
homogeneous  coi'irdinates  ./•://:  2 ://',  and  we  may  apph"  to  this 
plane  the  methods  and  formulas  of  three-dimensional  geometry 
with  quadriplanar  coi'irdinates. 

It  is  important  to  notice  the  connection  between  figures  in  the 
four-dimensional  space  and  their  intercepts  with  the  hvperplane  at 
infinity.  These  intercepts  we  shall  sometimes  call  fi'tii'i-n. 

The  e< piat  ion  (  •">  )  of  a  st  rai^ht  line  with  direct  ion  .  I  :/!:<':  /Mnay 
be  wi'itten  in  homogeneous  eoiirdinates  as 


:J7()  Fon;  IMMKNSIONAL  CKOMKTIIV 

\\  hence    it    appears    at    once    that    its   intercept    with    f  =  <>    is    the 
point    .1  :/::<':/>. 

The  eijiiatitui   of  a  hvperplane  is 

.i.r  +  /••//  +  <'z  +  />"'  +  i-:t  —  o, 

ami   its  trace  on  the  hyperplane  at    infinity  is  the  plane 

A.r  +  /.'//  -f-  ('2  +  /»/•  =  0. 
Similarly,  the  equations  of  a  plane  are 

J1.r  +  />t1.//  +  ri*  +  7Jlw  +  7::i/  ,-:0, 
•  I.,-''  +  /•'...'/  +  <  '.,z  +  />.,"•  +  /•-',/  =  <>, 
and  its  trace  on  the  hyperplane  at   infinity  is  the  straight  line 

.1  ,./•  +  /•',// +  rr?  +  /V/1  =  0, 
.I.r  4-  /•'.,//  4-  <'.,•  +  />.,"'  —  ^. 

A  Jiifj'H'rxplit'ri'  is  defined  as  the  locus  of  points  \\hosc  distances 
from  a  fixed  point  are  equal.  It  is  easv  to  show  from  ( '_' )  that 
the  equation  of  a  hvpersphere  is 

"„(  •''~+<r+^+  "'' )+  -  ",.'•' 4-  -  ".„'/'+  -  '<Jt+  -  fi4irt-\-,tf2=  0,    (  s  > 
and  that   its  intercept  with  the  hypcrplane  at  infinity  is  the  qiiailrie 

surfiu'°  .r+,r+r+fr^=n.  (0) 

This  surface,  ^'hich  we  call  the  ttf>x"}uti'.  plays  a  role  in  four- 
diniensional  sjfeonietry  analogous  to  that  played  l>y  the  inia^iiiary 
circle  at  infinity  in  three-dimensional  ^eoinetr\'.  All  hyperspheres 
contain  the  absolute.  The  hyperplane //'=  <>  intersects  the  absolute 
in  the  imaginary  circle  at  infinity  in  the  space  of  the  coordinates 
./•.  //.  ,:.  The  same  tiling  is  true  of  all  hyperplanes,  \\ith  the 
exception  of  the  minimum  hyperplanes,  to  he  considered  later. 

158.  Parallelism.  Any  two  of  the  configurations,  strai^lu  line, 
plane,  or  hyperplane,  are  said  to  lie  parallel  if  their  complete 
intersect  i<  in  is  at  inlinit  y. 

This  definition  ^'i\'es  us  nothing  new  concerning  parallel  lines. 
Fi>r  example',  we  have,  at  mice,  the  following  theorem: 

7.    77'  /'"  " '  //'  <'/<>/  1 1"  I  lit  i  n  x  i  n  1 1 -i    i /<  a  a  1 1  ft  i  n  i /It'  1 1  Hi'  I  >'  I  t'lllli  /  In  1 1   tl.i'i'if  1 1  in-. 

.\l'l/     t  It'll     III!  fill  1 1    I     1 1  III    X     //.        ///      ///.       HI  I  Illl'     Ill  ll  III       1 1  II'  I    ill    fl    r  III  I  III       I  111'      I  >ll  I  IIS. 


POINT  COORDINATES  371 

Neither  do  we  find  anything  new  concerning  ;i  line  parallel  to  a 
plane.  We  luive  already  seen  that  a  line  will  Mot  meet  a  plane 
unless  it  lies  in  the  same  hyperplane.  In  (In-  latter  case  the  line 
mav  interseet  the  plane  in  a  finite  point  or  he  parallel  to  it.  \Ve 
have  the  following  theorem: 

II.  If  <t  //'HI'  /x  jitini//''/  t'i  it  I'litni'  tJn1  tirn  //,•  in  (In'  »<tini'  liijjn  r- 
iihim-  it  nil  i/ifi/'/iinii'  f/i'if  h  i/jn  i'jil'1  in'.  Tlu'iiiii/li  iin//  [mi nt  tn  xixt/'t' 
//</,*  it  jn  nril  "/'  liiti-K  i>iiril//i'/  I"  it  tin  il  jilitiif. 

\Vlien  we  consider  parallel  planes  we  have  to  distinguish  two 
cast's.  Two  planes  are  said  to  he  rtmijilt'ti'lt/  /><ir«//i-/  it  they  in- 
tersect in  a  line  at  intinity.  and  are  said  to  he  x///////y  i><irnllfl 
if  they  intersect  in  a  single  point  at  intinitv  and  in  no  other 
point. 

From  theorem   XI,  (4).  ^  l.Vi,  we  have,  at   once,  the  theorem: 

///.   //'   tn'n     i/d 


In  fact,  completely  parallel  planes  are  the  parallel  planes  of  the 
ordinary  three-dimensional  geometry.  On  the  other  hand,  two 
simply  parallel  planes  do  not  lie  in  the  same  hvperplane  and  con- 
seijueiitK  cannot  appear  in  three-dimensional  geometry.  A  dis- 
tinction hetween  completely  and  simply  parallel  planes  is  brought 
out  in  the  following  theorem  : 

IV.     If  t  il'n    filitni'K    lire    i-nin  i>li'ti'l  I/   jut  I'ltl  li'l .    Hit  I/  lilli'   if  ii, 1C   /X    IHiralli'l 

In  xmitt'  Inn  if  tin-  nt/nr  itml,  in  j'lli'f,  tn  it  //»//<•//  nf  Ian  a.  If  tirn 
l>/in/i  x  <//v  xi in/ill/  jut /'iillil ,  tlni'i  IK  n  UHtifUf  i//r< '•//»//  in  iiii'//  ji!<titi' 
Kiii-h  tlnit  liin'K  with  tlnit  il  i  I'l'i-t'mn  in  litlnr  i>htn<'  <//v  fnti'nlh'1  tn  /I/UK 
it'tth  tin1  Kii/in  tlii'i 'i-f/ii/i  in  lli,'  nf//i/\  I'l't  htiiK  ii'itli  ii/ii/  "tin/'  i/n'ii't/'in 

III     "IK'     fililm'     Hl'l'     /Hl/-'llli  I     tn     ,/,,     Hill  'X     I  if    till       "till   l\ 

To  understand  thi>  theoi-em  note  that  if  t  \\  o  eompletely  jtaiallel 
planes  intersect  in  the  line  /  at  infinity,  any  line  in  one  plane  \\ill 
meet  /  in  .some  point  /'.  and  any  line  through  /'  in  the  >econd 
plane  will  he  parallel  to  the  first  plane.  If,  however,  t\\o  simph 
parallel  planes  intersect  in  a  single  point  /'  at  inlinitv.  the  only 
lilies  in  the  t  \\  1 1  planes  which  are  parallel  are  those  which  intersect 
in  /'.  It  may  he  noticed  that  this  properu  of  a  unique  direction 


?)~'2  KoriJ-niMKNSIONAL  CKOMKTKY 

is  found  also  in  two  intersecting  planes,  the  unique  direction  being 
that   of  the   line  of  intersection. 

A  plane  is  parallel  to  a  hyperplane  if  they  intersect  in  a  straight 
line  at  infinity.  Let  this  line  be  /.  Then  any  line  in  the  plane 
meets  /  in  a  point  /',  and  a  bundle  of  lines  may  be  drawn  in 
the  hyperplane  through  /'.  Then  each  line  of  the  bundle  is  par- 
allel to  the  given  line.  The  hyperplane  meets  the  plane  at  infinity 
in  a  plane  ///.  in  which  the  line  /  lies.  Any  plane  in  the  hyperplane 
intersects  ///  in  a  line  /',  which  has  at  least  one  point  in  common 
\\ith  /  but  which  may  coincide  with  /.  From  these  considerations 
we  state  the  theorem  : 

V.  If  <i  //l<nii'  itii'l  (i  hj/perjj/diii'  arc  punillt'1,  «>ij/  lin<'  in  f/n'  [thine 
ix  n<t/-'i//i'/  to  ,ii<'h  Inii'  of  ,i  on  mltf  in  tin1  lii/pt'i'jiltiHt',  (i nil  inn/  j>l<tne 

'di'dllcl  to  tin'  ///I't'ii   itldtlf. 


Two  hyperplanes  are  parallel  if  they  intersect  in  the  same  plane 
at  infinity.  Let  that  plane  be  m.  Any  plane  in  one  hyperplane 
meets  in  in  a  straight  line  /,  and  through  /  may  be  passed  a  pencil 
of  planes  in  the  other  hyperplane.  Again,  consider  any  two  planes, 
(Hie  in  each  of  the  hvnerplanes.  They  meet  in  in  two  lines,  /and  /', 
which  intersect  in  a  point  unless  they  coincide.  The  two  planes 
can  have  no  other  point  in  common  unless  they  are  in  the  same 
hvperplaiie.  lleiice  we  have  the  theorem: 

VI.  If  in'''  /ti//n'/'}>liitii-x  tin1  [xtriiUt'I.  i(n i/  fi/iiin-  <>f  one  I'K  I'vttiulctt'li/ 
iiitridft'l  /"  *"//(•'  jiliiin'  iin<l  1ti')i<'t'  t»  n  />t'/ii-//  i>t'  fifitttrx  "/'  ///c  of/n  r, 
iinil  <ni  if  jilitin'  <it  o/n-  /*  ifitttjilt/  i'ili'itlli'1  to  (i n  if  italic  H'fidtt'l't't'  of 

tin-    "///('/'    to    ?/'/(/'•//    //    /.s1    //"/    roiiifilitt'll/   /itlt'tlf/t'l. 

'I'he  anah'tic  conditions  for  parallelism  are  easily  given.  The 
necessary  and  sut'ticicnt  condition  that  two  lines  with  the  directions 


mid   lie  parallel   is   that    .  I,:  /.', :  <\ :  1\  --•  .  I .,:  // .,:  f .,:  I> ... 

Also  the  necessar\  and  siitliciciit  condition  that  two  hyperplanes 

.i/  +  /;..//  i-  <\?  +  />//'  +  /•:/  =  o 
I  .!.,./•  4-  r,  :i  f  c 2  f  />,//'  +  /•.'/  =  <J 

•  nld    be   parallel    is  that   .).:  /-'  :('  :  />         I  :  />'.:  ' '.,:  1> , 


POINT   COORDINATES 


Since  two  planes  are  simply  parallel  when  they  intprsect  in  a 
single  point  at  inlinity,  the  necessary  and  .sul'licicnt  condition  that 
the  two  philips  (.  4- r  i  ii  i 

{/','•      /'         <•        if,,      /•'/" o  J  (1  } 

an(i  f  /';// + \  •'! h  /y h  /y  ( -  * 

should  be  simply  parallel  is  that 


but  that  not  all  tlu-  otlu-r  t'oiirth-ordt-r  determinants  of  thp  matrix 


A     n     c     it     i-: 

1  1444 

sliould  vanish. 

That  the  t\yo  planes  (1  )  and  ('1  )  should  lip  completely  parallel 
their  traces  on  the  hyperplane  at  inlinity  must  coincide.  No\v  the 
determinants  of  the  matrix 


are  I'liickcr  coordinates  for  the  tract-  of  the  plane.  Therefore  the 
necessary  and  sutlicielit  condition  that  the  t  \\  o  planes  <  1  )  and  <  '1  ) 
sliould  he  parallel  is  that  the  determinants  of  the  matrix 


.should  hayp  a  constant   ratio  to  the  porrpspondint^  determinants  of 
the  matrix 


159.  Perpendicularity.  In  accordance  with  ( 'i  ).  \  \~>~,  t\\o  line 
with  the  directions  A-.H'J'-.I)  and  .\:l'.,:<':l>  are  said  to  li 
[lerjiendieu  lar  \\  hen 

.1  .1.  f  /;  /,'  ]  c  r   i  />  />   -   it.  (  1 


374  FOUR    DIMENSIONAL  GEOMETRY 

This  condition  may  be  ^iveii  a  useful  interpretation  in  the  hyper- 
plane  at  inlinity.  The  polar  plane  of  a  point  .1^:  //}:  z}:  /r^  in  the 
hyperplane  t  =  <K  \\'ith  respect  to  the  absolute  .r+  //"+  /J  +  i>'~  =  0,  is 
./•./•-(-//  //  -f-  ~  z  -f-  n'X'  =  I >.  Fijuation  (1)  therefore  shows  that  two 
perpendicular  lines  meet  the  hyperplane  at  inlinity  in  two  points, 
each  of  which  is  on  the  polar  plane  of  the  other  with  respect  to 
the  absolute.  Or.  otherwise  expressed,  tin'  nci-cxxnn/  <<nil  sufficient 
i-o/tijitinn  t/nif  tu'n  lini'x  ii/-i'  in  r/>i'ni//'i-/i/iir  i*  that  tlu'ir  tritrcx  mi  tin1 

1  I 

iii'intx  in  tchii'/i  t/tf  lini'  <'<>n)iet'thi//  t/ic  trnn'x  nn-i'tx  tin'  ulmnlutt'. 

A  line  is  said  to  be  perpendicular  to  a  hyperplane  when  it  is 
perpendicular  to  every  line  in  the  hvperplane.  For  this  to  happen 
it  is  necessary  and  sufticicnt  that  the  hyperplane  meet  the  hvper- 
plane at  inlinity  in  the  polar  plane  of  the  trace  of  the  line.  From 
this  follows  at  once  the  theorem: 

/.  Tln'i'iti/fi  tin '/  [>otnt  i /fin'/'  tn  "T  K'tfl/oitt  it  li  i/ [n'l'i'Iit  ni'  »ni'  it  ml 
nit!//  nth'  #ti'(lii/ht  Itni'  <'<in  /'C  ili'itn'n  I'l'/'jn'ml K'ulii r  tn  tin1  It //i/t'/'/i/itnr  ; 
dint  J/'niii  'till/  [mint  in  '//•  i('ttfi<>lit  it  xtt'itli/rtt  ftiti'  <>n<'  tlttil  o)ilif  oiif 

Since  in  the  plane  at  inlinity  the  polar  plane  with  respect  to  the 
absolute  of  the  point  A :  /•':  < ':  I>  is  the  plane  .!./•  +  /.'//  +  ( '•  -f  l>tc  —  0, 
we  have  the  theorem  : 

II.  .  1  ////  ////<•  /'i'/-/'i'/i'//i-i//iir  tn  tin-  }<///„,•/>!« a,' .  LI-+  /;//+ 1  '•  +  i>n<-\-  /•;=  0 

/nix  tin-  i/iri'i-t i'i/i  A:  /•':  < ':  I>.   <tn<l  '•<-///> /-.s,///. 

Any  three  lines  of  a  hvperplane  which  are  not  coplanar,  and  no 
two  ot  which  arc  parallel,  deteriiuiie  three  nonrollincar  points  ot 
the  trace  of  the  h\'perplaiie  at  inlinitv.  The  line  perjiendirnlar  to 
these  three  lilies  passes  through  the  pole  of  the  plane  determined 
by  the  three  points.  Consequently  we  have  the  theorem: 

///.   .1   ////'•  jn /-jiiii'l i'-nlii r   tn   th i->i    /inift  <>f  n   /////"  /'['linn'   ii'Jili'Ji   it/'i' 

In  particular  the  three  lines  mav  intersect  in  the  same  point. 
( 'i  mscfpient  ly  we  have  the  theorem: 

IV.   .1    1 1  ni',   ni'ii/  /«•    ilnnrn   j»  i'i«  Hilii-nliir   to    fj/i'i-i'    /tniK   i  ntiTfi'i'tniil 

ill     'I     ii'iiiit    ?o-lf     not     in     tli,     Kilnii'    liliDli.    ill/'/    if    /••>'    ///'//    /n  I'jii  Hit I'-lllil I'    tn 

f/n    In//"  r/i/'in,    1/1  ti  riiii/n  >l  lot  tin    t/ir<  i'  lin>*. 


POINT  COORDINATES  :}7~) 

A  line  is  perpendicular  to  a  plane  if  it  is  perpendicular  to  every 
line  in  that  plane.  From  this  we  have  the  theorem: 

V.  It'  it  Inn-  ix  perpendicular  tn  it  1ii/perpl<inf,  it   t'x  perpt'iidicuhir 

fn  (•/'(•/•//  jiliini'  tn  tlii-  Ii  i/pi' i'pl<i  tie. 

The  definition  of  perpendicularity  of  line  and  plane  is  the  same 
as  in  three-dimensional  geometry.  The  theorem,  however,  that  from 
a  point  in  a  plane  only  one  line  can  IK-  drawn  perpendicular  to  it  is 
no  longer  true. 

In  fact,  consider  a  plane  /  and  any  point  /'  in  it,  and  let  the 
tract-  of  /  on  /  .  l)  be  the  line  I..  Further,  let  //  be  the  conjugate 
polar  line  of  /.  with  respect  to  the  absolute  (  sj  (.lii).  Then  any  point 
on  /.'  is  the  harmonic  conjugate  of  any  point  on  I.,  ilence  any  two 
lines,  one  of  which  intersects  /,  and  the  other  //,  are  perpendicular. 
From  /'a  pencil  of  lines  may  be  drawn  to  meet  /.'.  Therefore  we 
have  the  theorem  : 

VI.  All  Uiu'x pei'i/end'n-nlar  tn  <t  plane  nt  it  fi.red  [mint  I!,-  in  a  plane. 

Tin'    tirn     i,/,l,,i  X     ill','     xllelt     fluff     Cl'crif    ////c     nf    n,u'     IX    perpendicular     tn 
,  r,  /'//   ////,•   nf  the    ntln  r. 

'Ihcse  planes  are  said  to  be  enniplitel  i/  peri>endic>diir.  Obviously 
they  do  not  exist  in  ordinary  three-dimensional  space. 

The  point  /'considered  above  need  not  lie  in  the  plane/.  Hence 
we  have  the  more  general  theorem: 

VII.  TJl/'nll</li    it  ml   point    nf  xpace    n//e   plane,    itntl    <>/////    mie,    eitn    It' 
pitxxi-d    completely   perpendicular    tn    ,/    i/ie,  it    plane. 

\\'ith  the  same  notation  as  before  let  /  be  a  ^iven  plane,  /'  a 
point  which  mav  or  may  not  lie  in  /,  and  /'.I  a  line  perpendicular 
to  /.  where  .1  lies  on  /.'.  Through  /'./  pass  a  plane  ///  intersecting 
/  <t  in  a  line  .17  through  .1.  If  .17'  is  the  conjugate  polar  of  .17, 
.17'  intersects  /.  in  a  point  /,'.  1>\  the  theory  of  conjugate  polar  lines. 
Thru  if  <t>  is  an\-  [loint  of  /.  the  line  >,>/!  lies  in  /  and  is  perpen- 
dicular to  ///.  Therefore  we  have  the  following  theorem: 

VIII.  It   ii   jiliii/i    ni   cniitiiinx   it   line   /"'/'/" 'iid icnlit r  tn  ii   fi/i/ne  /,   ///• 
filiin,    I   eniitiinix   ii    line  /"/'/"  n<l ii'i'l'tr   tn   ///. 

T\\o  planes  such  that  each  contains  a  line  perpendicular  to  the 
other  \\  e  shall  call  xenii per/iendieuliir  planes. 


370  FOI'R   DIMKNSIONAL  C  KOM  KTl;  V 

From  ilu-  foregoing  we  easily  deduct'  the  following  theorem: 

IX.  Tin'    in'1'fxxiii'il  iiinl  Nltffifit'ltt  f«»<//ti»//  f/nif  tti'n  jiftith'X  xlnnilil  fn' 
Ki-iiiiiii'i'fit'nilt<'nl<ii'  ix  tlnit  tin'  tniff  lit  infinity  •>>'  I'itln'i'  xhtialil  inta'xt't't 

in  "Hi"  ji'iiiit  th,'  fiiij i/i/iitf  I'lilnr  i/'ith  ri't/ii'i-f  t"  tin-  tiliKii/Hti1  <>t'  tin-  //'tiff 
i,t'  tit,  "tin  r.  Tin'  nffi'ssiirif  (tint  xntjifi,//f  fntnlitnni  (lint  t/r»  filiiin-x 
x/tuti/il  />,'  i-ninplt'ttiif  jH'rjH'Htlii'iilitt'  IN  tlnit  tin'  tr«ff  of  fit/n't'  nJumhl 

In1  tin'  fiiij iii/ntf  j><i/(<r  nt'  tin'  tfiicf  nj   tin'  otl/cr. 

It  two  st'iitipt-rpcndiciihir  plain's  lie  in  the  same  hyperpliuu1, 
thev  intersect  in  a  line  and  are  the  ordinary  jierpendicnlar  planes 
of  three-dimensional  t^eometrv.  If  two  seiniperpendicular  planes 
are  not  in  the  same  hvperplane,  thev  intersect  in  a  single  point.  It 
this  point  is  at  infinity,  the  two  planes  are  also  simplv  pai'allel. 
In  these  crises  the  traces  I.  and  M  intersect  in  a  [mint  <\  which  is 
harmonic  conjugate  to  both  A  and  //.  From  this  follows  the 
theorem  : 

X.  Tn'"    wtnivi'rpt'nilit'iilur   jrfmn'H    nntit   In1    tdnijilif   i«trnUi-l.     Tin' 
</i/-fffi"tt   i if  tin'   /"d-K/lfl  Uin-x  i f  tin'   t H'n   ji/n/tfx   /.v   tin- n   <>rt lt<i<i<nntl  t" 
tin'    lUl't'l'tioHIS    "J    tin'   l>f/'l>f/ti/t''H/i(r    //itfx. 

It  is  to  he  noticed  that  in  this  case  the  direction  of  the  pai'allel 
lines  is  similar  to  that  ot  the  line  of  intersection  of  sennperpen- 
diciilar  planes  in  the  same  hyperplane. 

A  plane  /  is  perpendicular  to  a  hyperplane  //  when  it  contains 
a  normal  line  to  the  hyperplane.  The  trace  I.  of  the  plane  then 
passes  through  the  pole  of  the  trace  //  of  the  hyperplane,  and  the 
conjugate  polar  //  of  I.  lies  in  //.  Therefore: 

XI.  It   ii  />/ii /if  /.s-  /if/'/if/K/ifii/itr  f"  ii    fitf/H'/'ji/iini1,   it   i'x   f"in/il,'t,'l i/ 
/"•/'/"  inlifiil<i  r  tu  ,  -iii-lt  jiliiin  >f  <i  jn-iifil  if  jKiriilli  I  /'/it/ifx  'it'  tin-  Ii  if/if  r- 
liliiin-  iiinl  xfiiiifn  I'IH  tnltfiilnr  t"  ,1-,'rn  i>f //,•/•  filiini'  "t'  tin'  luijii'i-jilii in-. 

The  angle  hetweeii  two  livperphincs  mav  he  dclined  as  the  angle 
between  their  normal  lines.  Ileiice  two  hvperplanes, 

.(..'•  f  /•',//  f  '\'-  +  /V  +  ^-V  ° 

and  A  ./   f  //,//  -f  (' .;  -f  />//•  +  E  >  0, 

are   jierjieiidiciilar  \\hen   and   onh'  \\heii 

.1.1     f  /.'/•',  +  ('.('    f  l>  l>  0. 


POINT  t'OfWDIXATKS  377 

This  is  the  condition  that  the  tnu-cs  at  infinity  of  the  two  hyper- 
planes ure  such  thut  each  contains  the  pole  of  the  other,  as  might 
he  inferred  from  the  definition.  From  this  \ve  have  the  theorems: 

XII.  It  t  n'o  ht/perplanfx  <i/'i'  pi'/'pt  '  ml  ii-iil  iir,  the  normal  f»  eith>  r  1  i'»tn 
(tiii/  i>nint  <>t  tliftr  intersection  //<•*  ///  tin'  other. 

XIII.  Ant/  hyperplane  paxxi'il  thmuah   a   normal  to  another  hj/per- 
pltine   ix  perpendicular  t<>  t/iat  Jiyperplane. 

Since  in  t  —  ()  the  intersection  of  two  planes  is  the  conjugate 
polar  of  the  line  connecting  the  poles  of  the  planes,  we  have  the 
theorem  : 

XIV.  TJte  plane  <>f  inter  sect  tun  of  (tea  perpendicular  hyperplane*  i* 
evinplt'tely  pt'i'pi'tnliL'uhir  to  <t/i//  plant1  ileterinined  l>t/  tic<>  t/ttcr'Nt-fttnij 
nvrmalx  to  //<»•  hyprrplant'8. 

In  tlnj  hvpcrplane  at  in  I'm  it  v  we  may,  in  tin  infinite  number  of 
wavs,  select  a  tetrahedron  AH('I)  which  shall  he  self-conjugate  with 
respect  to  the  absolute.  From  any  finite  point  <>  draw  the  lines 
<>.l,  (>!;.<)('.  <>/>.  \\V  have  a  configuration,  the  properties  of  which 
are  given  in  the  following  theorem  : 

XV.  From  auij  jioint  in  xpai-e  nuiij  of  drawn*  in  a/i  infinite  number 
of  trail*,  foi/r  inuttuillii  perpendicular  line*,      h't't't'//  tln'ee  of  these  lin>'x 
Jetet'i/ttnex  <i  Jti/pet'plitne  perpendicular  to  the  //_///>(•/•/'/<//«'  determined 
liil  iin//  other  three.     Every  pair  <>t  the  linex  determines  a  plane  which 
i*  completely  perpendicular  to  tlmt  determined  /<//  thf  other  pair  of 

the    linex, 

A  sjiecial  case  of  the  configuration  described  above  is  that  formed 
by  the  coordinate  hyperplanes  A  =  U,  }'  -—  0,  7,  =  0,  \Y  --  0. 

1»\  (  ti  ),  ^  Io7,  the  cosines  of  the  angles  made  with  the  coordi- 
nate hyperplanes  by  the  livpcrplane 

"'  +  K=  0 


\  .  /-  +  /;•  -f  (  '-  +  />-          \  .  i  -  +  /'•  -f  '  '' 

when  ./-  f  /;-+  r-  f-  //-  i-  U. 


378  R)UR-I)IMEXSIOXAL  GEOMETRY 

\Yc   may  denote  these  by  /,  >n,  //,  /•  respectively,  and  write  the 
(•([nation  of  the  hyperplane  in  the  form 

fa  4  my  4  HZ  +  rw  4-  />  =  0, 

with  /" 4- //r 4- >r 4- >""  =  !•  The  equation  is  then  in  the  normal  form, 
and  it  is  easy  to  show  that  }>  is  the  length  of  the  perpendicular 
from  the  origin  to  the  plane.  Also  by  the  same  methods  as  in 
three-dimensional  geometry  we  may  show  that  the  length  of  the 
perpendicular  from  any  point  (r  ,  // ,  z{,  tc  )  is  fa -\-iny  +nz+rw-\-j). 
Let  us  now  take  any  configuration  described  in  theorem  XV, 
and,  writing  the  equation  of  each  of  the  four  hvperplanes  in  the 
normal  form,  make  the  transformation  of  coordinates  given  by  the 
equations  in  nonhomogeneous  coordinates: 

j-'  =  //  4-  ///,//  4-  »r7  +  ri"'  +  /'i' 


//•'  =  /vr  4-  /"4//  4-  /'/  4-  /V'1  +  /'4' 
with  the  conditions         /-'4-  ///j4-  /<:  4-  r'~  =  1, 

//j.4-  '","li.  +  tl,fli-^~  r,'V  =  "•          ( '  ^  k} 

The  new  coi'irdinates  are  the  distances  from  four  orthogonal  hyper- 
planes,  and,  in  fact,  our  discussion  shows  that  the  same  is  true  of 
the  original  coordinates. 

In  the  new  system  the  equation  for  distance  is  unaltered,  namely, 

'/  =  v'(  j\,  -  .'•;  r  4-  < !/',  -  //;  >3 + <  *a  —  *i  >'• + <  "•',  -  n'\  )•, 

and  if  we  place  //•'— 0  we  have  the  ordinary  Euclidean  geometry 
in  three  dimensions.  This  justilies  the  statement  already  made  in 
anticipation,  which  we  now  t^ive  as  a  theorem: 

XVI.  In  j'<iit/'-(li//ti'iixlnti'il  Kiirliilt'dii  x/nii-i'  t/it'  i/i'iinii'f/'if  iii  (in>i 
/ti//«r/>/<i/i<\  fuf  ii<hi'-li  .\--\-  /:-+  C~+  //-—  0,  IK  ttmt  <>f  (/„'  tixttnl 
thri'i'-iltint'tlxiumil  I'liii'I nli'H n  </fi>un't ri/. 

160.  Minimum  lines,  planes,  and  hyperplanes.  In  the  discussion 
of  the  previous  section  we  have  had  to  make  exception  of  the  rases 
in  which  the  direction  quantities  ./,  /.',  < ',  />  satisfy  the  condition 

J -+/'''-' 4- ''- 4- />'-=0.  (1) 


I'OINT  COORDINATES  370 

We  shall  now  examine  the  exceptional  cases. 

Obviously  the  necessary  and  sufficient  condition  that  the  direction 
quantities  of  a  straight  line  satisfy  equation  (  1  )  is  that  the  line  inter- 
sects the  absolute,  or,  in  other  words,  that  the  trace  at  infinity  of  the 
line  lies  on  the  absolute.  The  necessary  and  sufficient  condition  that 
the  quantities  J,  />',  < '.  I >  in  an  equation  of  a  hyperplane  satisfy  (  1  )  is 
that  the  trace  at  infinity  of  the  hyperplane  is  tangent  to  the  absolute. 
In  this  case  the  hvperplane  is  said  to  be  tangent  to  the  absolute. 

The  straight  lines  which  intersect  the  absolute  are  the  minimum 
lines  of  three-dimensional  geometry. 

In  fact,  the  hyperplane  //'  =  <),  which  by  theorem  XVI.  ^  l-~>(.», 
represents  any  ordinary  hyperplane,  meets  the  absolute  in  the  imag- 
inary circle  at  infinity,  and  the  lines  in  the  hvperplane  which  meet 
the  absolute  are  therefore  the  minimum  lines  of  the  hyperplane. 
Also,  if  any  line  meets  the  absolute  in  a  point  /'.  a  hyperplane 
can  evidently  be  determined  in  an  infinite  number  of  ways  so  as 
to  contain  the  line  and  not  be  tangent  to  the  absolute.  We  have, 
therefore,  nothing  new  to  add  to  the  three-dimensional  properties 
of  minimum  lines. 

In  four-dimensional  space  there  go  through  every  point  -f.  ~  mini- 
mum lines,  one  to  each  of  the  points  of  the  absolute.  These  lines 
form  a  hypercone.  A  hyperplane  through  the  vertex  intersects  the 
hvpeivone  in  general  in  an  ordinary  cone  of  minimum  lines,  and  a 
plane  through  the  vertex  intersects  the  hypercone  in  general  in  two 

('onsider  now  any  plane.  Its  trace  in  the  hyperplane  at  infinity 
is  a  straight  line  which  may  have  any  one  of  three  relations  to 
the  absolute:  (  1  )  it  may  intersect  the  absolute  in  two  distinct 
points:  cl)  it  may  be  tangent  to  the  absolute:  ('•'•>)  it  may  lie 
entirely  on  the  absolute. 

The  first  case  is  the  ordinary  plane,  the  second  the 
plane  of  three-dimensional  geometry.  In  fact,  if  an\ 
character  (  1  )  or  (  '1  )  is  given,  it  is  clearly  possible  to  tin 
plane  which  will  contain  it  and  not  be  tangent  to  the  absolute, 
The  ordinary  plane  is  characterized  hv  the  property  that  through 
any  point  of  n  <n>  two  minimum  lines,  and  the  minimum  plane  of 

I  r"l  I 

three-dimensional    type    by   the    property  that    through   every  point 

of     it     'roeS    OIK 


;} SO  K< ) I"  1 1  - 1 ) I M  KN SI(  )X A L  (1  K(  >M KT 1  { V 

The  third  type  of  plane  is,  however,  not  found  iu  the  ordinary 
three-dimensional  geometry.  For  if  a  plane  meets  the  absolute  in 
a  straight  line,  any  hyperplane  containing  it  contains  this  line  and 
therefore  intersects  the  absolute  in  two  straight  lines.  The  geometry 
in  this  hyperplane  is  therefore  a  geometry  in  which  the  imaginary 
circle  at  infinity  is  replaced  by  two  intersecting  straight  lines.  Its 
properties  will  therefore  differ  from  those  of  Kuclidean  space. 

A  plane  at  infinity  intersecting  the  absolute  in  two  straight  lines 
is  tangent  to  it.  Therefore  a  plane  of  the  third  type  lies  only  in 
hvperplanes  tangent  to  the  absolute.  A  unique  property  of  these 
planes  is  that  any  straight  line  in  them  meets  the  absolute  and  is 
therefore  a  minimum  line.  In  other  words,  the  distance  between 
any  two  points  on  planes  of  this  type  is  x.ero.  \Ve  shall  refer  to  a 
plane  of  this  type  as  a  minimum  jilnnf  <>f  t/n-  xi'mml  kin<L 

Consider  now  a  hyperplane  which  is  tangent  to  the  absolute. 
The  equation  of  such  a  hyperplane  is 

AJT  +  />'//  +  Cz  +  !»'>  +  K  =  0 

with  A~+  />'"+  f'~+  I>2=  0.  From  analogy  to  three-dimensional 
geometry  we  shall  call  such  a  hyperplane  a  minimum  Jii/perplane, 
It  has  already  been  remarked  that  in  a  minimum  hyperplane  we 
have  at  infinity  two  intersecting  straight  lines  instead  of  an  imagi- 
nary circle.  There  will  be  a  unique  direction  in  the  hyperplane; 
namely,  that  toward  the  point  of  intersection  of  the  two  imagi- 
nary lines  at  infinity.  For  convenience  we  shall  call  a  line  with 
this  direction  an  n.rfx  of  the  hyperplane. 

Through  every  point  of  the  hvperplane  goes  an  axis,  and  through 
every  axis  go  two  minimum  planes  of  the  second  kind,  each  con- 
taining one  of  the  two  intersecting  lines  at  infinity.  Any  other 
plane  through  the  axis  is  an  ordinary  minimum  plane.  The  cone 
of  minimum  lines  through  a  point  splits  up,  then,  into  two  inter- 
secting planes. 

Any  plane  not  containing  the  axis  intersects  the  absolute  in  two 
distinct  points  and  is  therefore  an  ordinary  plane. 

Since  a  minimum  hvperplane  intersects  /  -  -  <>  in  a  plane  tangent 
to  the  absolute,  the  normal  to  the  hvperplane  passes  through  the 
point  of  tangeiiey,  which  is  the  point  of  intersection  of  the  two 
straight  lines  at  infinity.  Hence  the  axes  of  a  minimum,  hyperplane 


POINT  COORDINATES 


3S1 


are  the  normals  to  the  hyperplane.  The  axes  are  therefore  normal 
also  to  cverv  plane  in  the  minimum  hvperplane. 

Let  the  plane  of  the  li^nre  (  Fi^.  till  )  !«•  the  plane  of  intersection 
ot  a  minimum  hyperplaue  with  the  hyperplane  at  inlinitv,  and  let 
the  two  lines  <>,{  and  <>/!  be  ihe  intersection  of  the  plane  with  the 
absolute.  '1  hen,  it  /,  is  the  trace  ot  any  ordinarv  plane,  the  normal 
to  the  plane  passes  through  <>  and  is  an  axis  of  the  hvperplane. 
Two  ordinary  planes  in  the  minimum 

hvperplane,  therefore,  cannot  be  per-  /     r 

pendicular  to  each  other. 

But   consider   a   minimum   plane   of 
the  Iirst  kind  whose  trace  on  the  hvper-       \ 
plane  at  inlinitv  is  the  line  <><).    The 

conjugate  polar  of  the  line  <><,>  is  a  line      '_ 

<>!(.  Consequently  anv  two  minimum 
planes  of  the  iirst  kind  whose  traces  / 

are  < »t>  and  <>/,'  respectivelv  are  com- 
pletelv  perpendicular.  This  state  of 
two  completely  perpendicular  planes 

Ivin^  in  the  same  hvperplane  cannot  be  met  in  an  ordinary  hyper- 
plane and  is  therefore  not  found  in  Euclidean  geometry.  This 
is  due  to  the  fact  that  in  an  ordinarv  hyperplane  only  one  mini- 
mum plane  can  be  passed  through  a  minimum  line,  while  in  a 
minimum  hyperplane  a  pencil  of  minimum  planes  can  be  passed 
through  an  axis  of  the  hvperplane,  and  these  planes  are  paired 
into  completely  perpendicular  planes. 

Finally,  it  mav  be  remarked  that  a  minimum  plane  of  the  second 
kind  is,  in  a  SCUM-,  eompletelv  perpendicular  to  itself,  for  the  lines 
<>.!  and  <>!'>  are  each  self-conjugate. 

For  the  sake  of  an  analvtie  treatment  let  us  suppose  that  a 
minimum  hvperplane  has  the  equation  z  -  /"'  =  ",  and  let  us  make 
the  uonorthogoiuil  change  <>f  coordinates  expressed  hv  the  equations 

z'  ---  z  +  hi', 
a-'  ==  2       in'. 

Then    the   formula  for  distance   becomes 

<r~--  ( .'•„  —  .'•,  r  + 1  >f.,  -  '/, 'r  +  ( .?'  —  : •',  >  <  "•'  -  "•! ). 


FOUR-DIMENSIONAL  (iKoMKTKY 

In  the  hvperplane  //•'—<>  a  point  is  tixed  by  the  Coordinates 
a;  i/,  ~',  and  the  distance  between  tun  points  becomes 

•I  =  (•>',,—  •'',  >"  +  (//.,—  //!  )"• 
The  equation  of  the  two  straight   lines  at  infinity  is 

/'  +  >r  =  0, 

and  the  e<piations  of  anv  axis  of   the  hvperplane  is  ./•  —  ./•,  //  =  //,,. 
In    the    formula    for   distance    the    coordinate  z'  does   not    occur. 
Hence  the  distance  between  two  points  is  unaltered  Ity  displacing 
either  of  them  along  an  axis. 
Consider   the  equation 

(•''  -  •''„ )"  +  (.'/  -  .'/„)  =  ""• 

This  represents  the  locus  of  points  at  a  constant  distance  <>  from 
a  fixed  point  r(i,  //.^  .r.  where  .r  is  arhitrarv.  From  the  form  of  the 
equation  the  locus  is  a  cylinder  whose  elements  are  ;ixes.  Kvcrv 
point  on  the  cylinder  is  at  a  constant  distance  <i  from  each  point 
(>f  the  axis  ./•  =  ./•  ,  //  =  >/  . 

The  almvc  are  some  of  the  peculiar  pro|iertics  of  a  minimum 
hvperplane. 

161.  Hypersurfaces  of  second  order.    Consider  the  equation 

^\ ',./;/',=  0  (nti=  -/..  >  (1) 

in  the  homogeneous  coordinates  of  a  four-dimensional  space  in 
which  no  hvperplane  is  singled  out  to  In-  given  special  .significance 
as  the  hyperplane  at  infinity.  The  sp;icc  i>.  therctdrc.  a  projccti\c 
space.  The  student  will  have  no  difficulty  in  showing,  by  the  methods 
of  ;j  s^.  that  the  coi'u'dinates  may.  if  desired,  be  interpreted  a- 
equal  to  the  distances  from  live  hvpcrplanes.  each  distance  multi- 
plied bv  an  arbitrarv  constant.  However.  \\  e  shall  make  no  use  of 
this  propertv.  and  men t  ion  it  onlv  for  the  analo^ry  bet  ween  t  he  present 
coorilinates  and  ijuadriplanar  eoi'inlinates  in  three-dimensional  space. 
Filiation  (  1  >  represents  a  hypersurfacc  of  the  second  order.  It' 
//  and  :  are  anv  fixed  points,  the  line 

p.r  =  /ft+  \:  (  -2  ) 

intersects  the  hypersurface  in  LTciii-ral  in  t  \\'o  distinct  or  coincident 
points  or  lies  entirely  on  it.  Therefore  anv  hvperplane  intersects 
the  hvpc-rsurface  in  a  two-dimensional  extent  which  is  m,-t  bv  anv 


1'olNT  COORDINATES  ;{s:J 

line  in  two  points  and  is  therefore  a  quadric  surface,  or  else  tin1 
hyperplane  lies  entirely  on  tin-  hypersurlace.  Similarly,  any  plain' 
intersects  the  hvpersnrt'ace  (  1  )  in  a  conic  or  lies  entirely  on  it. 

Let  us  consider  these  intersections  more  carelullv.  It  in  equa- 
tion (  '_'  )  the  point  //,  is  taken  on  the  hypersurface,  the  line  will  meet 
the  li\  persurfaee  (  1  )  in  two  distinct  points  unless  the  donation 

5>*'/A=o  <:;> 

is  satisfied  l>v  the  point  zt.  In  the  latter  case  the  line  (  '2  )  meets 
(  1  )  in  two  points  coinciding  with  //,,  unless  also  zt  is  on  the  hyper- 
surface,  in  which  case  the  line  lies  entirely  on  the  h\  persurfaee. 

'This  means  that  if  //,  is  on  the  hvpersurlaee  (1  ),  any  point  on 
the  hyperplane  ''  =  °  4 


but  not  on  the  hvpersurface,  if  connected  with  //,,  determines  a 
straight  line  tangent  to  the  livpersurfaee,  and  this  property  is 
enjoyed  l>y  no  other  point.  Hence  the  hyperplaue  is  the  locus  of 
tangent  lines  at  //.  and  is  called  the  tuti'/ftif  fij/jn'rj>tttHf'. 

The  hvperplane  (  4  )  intersects  the  hypersurface  in  an  extent  of 
two  dimensions  which  has  the  property  that  any  point  on  it  deter- 
mines with  //(  a  line  entirely  on  it.  It  is  the  re  fore  a  cone  of  second 
order.  Therefore,  ////•"//////  >i/tf/  [>"int  of  thf  hi/fH'rxurfnfe  i/'x'it  <i  <-«/^' 
nt  xfi'titt/Jtt  /t//i'x  ///n/i/  1'iitii'ili/  nit  tlic  li  i/  JUT*!!  rt  <ii'i'. 

An  exception  to  the  alioye  occurs  when  f/  is  a  point  satisfying 
the  e(1uatio,,s  ^  +  ,,^  +  ^  ^+  n  ^  +  if^  =  0-  (-} 

Such  a  point,  if  it  exists,  is  a  sint/nliir  pm'nf.  At  a  singular  point 
the  ei|uatioii  of  the  tangent  hyperplane  becomes  illusive.  Any  line 
through  a  singular  [>oiiit  meets  the  li\  persurfaee  in  two  coincident 
points,  and  it  an\r  point  on  the  hypersurface  is  connected  with  the 
singular  point  by  a  straight  line,  the  line  lies  entirely  on  the  hyper- 
surlai'e.  I-.i  |  nat  ions  (  •>  )  do  not  always  liave  a  solution  :  but  it  they 
have,  the  solution  is  a  point  of  the  surface,  since  equation  (1  )  is 

111  Hill  '^enrolls. 

1  1  //,  is  any  point  .  whether  on  t  he  hvpersnrface  or  not,  equation  (  4  ) 
deli  i  icv  a  hvperplaiie  called  the  /m/iir  In//n  /•/  >/<i>/f  of  //  .  It  t  he  equal  ion 
ol  the  polar  hyperplane  is  \\rilieii  in  the  form 


/Kl  I-'nUMMMKNSION.YL  CKO.MKTKV 

I- rom   this   it    follows   that   anv  point    has  ;i  definite  polar  hvper- 
plane.    Tin'  converse  is  true,  ho\\ e\  er,  onlv  if  the  determinant 


docs  not  vanish.  The  vanishing  of  tliis  determinant  is  the  necessary 
and  sufficient  condition  that  equations  (.V)  should  have  a  solution. 
Therefore  we  sav  : 

If  <i  Jii/perpl'im' of  t/u1  ai'i-ond  ofili'r  tun*  n<>  Kinc/ulnr  points,  to  ft'iT// 
p»int  in  xy»/<v  fvrrcxptnitlx  <>  i//i/i/><f  /><i/<tr  Jii/perplctne,  <tnd  to  <T<T>/ 
liifpfrplnnc  w/v.*y>"/f<7*  n  n/i/i/tn'  /><//,>.  '/'//>•  ni't'efwury  and  sufficient  c<»t- 
<liti"tt  f<>r  ////x  ///  in-fur  /x  tloit  tin1  discriminant  <ii/c\  should  not  vatilsli. 

If  the  hvpersurfaee  lias  a  sin^nlai1  point,  it  is  easv  to  see  that 
I'very  polar  hyperplane  passes  through  that  point.  Therefore  onlv 
hvperplanes  through  the  singular  points  can  have  poles. 

The  properties  of  polar  livperphines  are  similar  to  those  of  polar 
planes  of  three-dimensional  ^eonx't  ry.  and  t  he  theorems  of  ij  It '2  niav, 
witli  slight  niodilicat  ions,  he  repeated  for  the  four  dimensions. 

AVc  mav  also  employ  some  of  the  methods  of  £  !>3  in  classi- 
f\  iii^'  hvpersnrfaees  of  the  second  oi'der.  Let  us  take  the  general 
case  in  which  no  singular  points  occur.  There  is  then  no  dif'ticultv 
in  applving  these  methods  to  show  that  the  equation  mav  he 

1T(IU(T(1     l"  ,T+,.;+,.^,.;+,.^,». 

The  cases  of  h\p|  icrsurfaccs  wit h  singular  points  are  more  tedious 
if  the  elemeiitiirv  methods  arc  used.  It  is  preferable  in  these  cases 
to  use  the  methods  of  elementary  divisors. 

162.  Duality  between  line  geometry  in  three  dimensions  and  point 
geometry  in  four  dimensions.  Since  the  straight  line  in  a  three- 
dimensional  space  is  determined  l>v  four  coordinates,  it  will  he 
dualistic  with  the  point  in  four  dimensions.  In  order  to  have 
coordinates  of  the  four-dimensional  space  which  arc  dualistic  with 
the  Klein  coordinates  of  the  straight  line,  we  will  introduce  hexa- 
spherical  coordinates  in  four-dimensional  space  analogous  to  the 
pent aspherieal  coordinates,  of  three-dimensional  space. 


POINT  r<  ><">i;i>lNATKS  :JS."> 

Fdllo\\ 'in^  tlic  analogy  of  ^  117.    1  ~'-\,  let   us  place 


p,\=lz 

p.r.=  -2H\ 

p.r6  =  /(  .V'-'  +  Y-+Z-+  II'2  +  1  ).  (  J  ) 

where  rf  +  r.r-f-  ./•.:  +  ./•;  +  ,/v  +  ./-,:  =  <». 

The  coordinates.;',  are  hexaspherica]  coordinates.  The  locus  at 
iniinitv  has  the  equation  j-  -f  /./-(,  =  0,  and  the  real  point  at  iniinitv 
has  the  coi'.nlinates  1:0:0:0:  0  :  /. 

The  e(|iiation 

is  that   of  the  hvpersphere 

There  an-  four  varieties  of  hyperspheres  : 

1.    Proper  hyperspheres,      ^  '/:—<>,  a  4-  ''',.  —  0. 

'2,    Proper  hyperplanes.         N  ,,~  =+ o^  ,f  . 

•  '>.    Point  hvperspheres,         'V/^:=0,  n  - 

\.    .Minimum  hvperplanes,   ^'',J  "  (l,  f/  +//^f=0. 

(  )n  the  other  hand,  we  mav  interpret  the  coi'irdinatcs  .?-(  as  Klein 
coordinates  of  a  straight  line  in  a  space  of  three  dimensions. 

For  convenience  we  will  denote  l>v  >'..  the  three-dimensional 
point  space  in  which  .?•.  are  line  coi'irdinati's,  and  l>v  ^i.  the  t'oiir- 
d miensii >nal  point  spaee  in  which  ./•,  are  hexaspherical  ctiJirdinates 
of  a  point.  Then  the  coordinates  1  :  <i  :  0  :  0  :  <>  :  /.  \\hieli  in  !i 
repi-eseiit  the  I'eal  point  at  inlinitv,  represent  in  ,s'  a  straight  line  /. 
wln'eli  has  no  peculiar  relation  to  the  line  space.  In  tact,  /acquires 
its  unique  significance  onlv  liccause  of  its  dualistie  relation  to  ^  . 
Also  the  equation  ./•  4- ''.'',,=  0,  which,  in  —  ,  rc]irc>ent<  the  hvper- 
plane  at  iniinitv,  represents  in  N  a  special  line  complex  •-.  o|  which 
the  line  /  is  the  axis.  With  these  preliminar\  remarks  \\  e  mav 
e.xhihit  in  parallel  columns  the  relation  Ketxvecii  .s'.  and  1(. 


For  K- 1 >IM  KNSION  A  L  ( }  KOM  ETM  V 
2  >' 


Point. 

Real  point  at  infinity. 

Proper  hypersphere. 


Lino. 
Line  /. 

N  on  special  line  complex  not  con- 
taining /. 

Nonspecial  complex  containing/. 
Special  complex  not  con  taming  /. 
Axis  of  special  complex, 
Special  complex  containing  /. 
Special  complex  f  with  axis  /. 


Proper  hvperplane. 

Point  liy  jiersphere. 

('enter  of  point  h  vpersphere. 

"Minimum  hyperplane. 

1  1  vperplane  at  intinity. 

Two  points  on   same  minimum  Two  intersecting  lines. 

line. 

Any  imaginary  point  at  intinity.  Line  intersecting  /. 

Points   common    to  t\vo  hvper-  Line  congruence. 

spheres. 

Vertices    of    two    point     hyper-  Axes  of  line  congruence. 

spheres    which    pass    through    the 
intersections  of  two  hvperspheres. 

Circle  defined  by  the  intersect  ion  Regulus. 

of  three  liy  persplieres. 

Two  circles  such  that  cadi  point  Two  re^uli  generating  the  same 

of  one  is  the  center  of  a  point  hyper-      qiiadrie  surface. 
spliei-e  passing  through  the  otlicr. 

The  use  of  liexasplierical  coordinates  gives  a  four-dimensional 
space  in  which  the  ideal  elements  differ  from  those  introduced  liv 
the  use  of  ('artesian  coordinates,  as  lias  been  explained  in  ^  \'2'-\. 
Such  a  space  is  in  a  one-to-one  relation  with  the  manifold  of  straight 
lines  in  S  . 

It  \vc  wish  tn  retain  in  ii  the  ideal  (dements  of  the  ('artesian 
geometry,  the  relation  between  A',  and  ~(  ceases  to  be  one-to-one  for 
certain  exceptional  elements.  T<>  show  this  we  will  niudifv  c(|na- 
tions  (  1  )  bv  introducing  homogeneous  coordinates  in  —  and  have 

PJ\    •-  .'•'+  <r  f  r-f  "'"—  f", 

p.r         '2  ./'/. 
pjr        'I  >/t, 


p.i'.       -  "'f. 

p.t\.  =  i  (  .r  4-  //-  4-  r  4-  ii'-  + 


POINT  COORDINATES  387 

If  we  use  these  equations  to  establish  the  relation   between  the 

lines  nf  .s'  and  the  points  of  ^,  we  shall  have  the  same  iv>ults 
as  before,  with  the  following  exceptions,  all  ol  which  relate  to  the 
ideal  elements  of  ^4.  Anv  point  in  i4  on  the  hyperplane  at  inlinitv, 
but  not  on  the  absolute,  corresponds  to  the  line  /:  and  the  line  / 
corresponds  to  all  points  on  /  —  n,  but  not  on  the  absolute. 

Anv  point  on  the  absolute  corresponds  to  a  line  in  .V  which  at 
tirst  sin'lit  seems  entirely  indeterniinate,  but  if  we.  write  equations 
(  o  )  in  the  form 


it   appears  that  a  point   on    the   absolute   corresponds  to  a   line   for 
wllirh  V-'V,=  1  :<',  .V.r;:.V.r  =  ,•://:-//•. 

This  is  a  one-dimensional  extent  of  lines.  One  line  of  the  extent 
is  alwavs  /,  and  another  is  I  :  j-  :  i/  :  z  :  tr  :  i.  The  general  line  mav 
be  written  as  (  1  4-  X  ):  ./•  :  //  :  :  :  ir  :  /(  1  -f  X  ).  I>y  ^  1  '-\\  the  extent  is, 
therefore,  a  pencil  containing  /.  Then,  to  any  point  on  the  al)solute 
corresponds  anv  line  of  a  certain  pencil  containing  /. 

It  is  easv  to  show  that  auv  line  \\hich  intersects  /  correspoiuls 
to  a  definite  point  on  the  absolute. 

It  is,  of  course,  possible  to  interpret  equation  t  =  0  in  equations  (  '•]) 
as  the  equation  of  anv  hvperplane  in  a  projective  space  with  the 
coordinates  -/'://:  ?  :  ""•  /.  The  absolute  is  then  replaced  bv  a  (piadric 
surface  <l>  in  the  hvperplane  t  =  ".  The  correspondence  between 
A',  and  ^(  is  then  less  sjiecial  than  the  one  we  have  considered. 

EXERCISES 


'2.  1'ctiiie  inversion  with  respect  to  a  li\  perspheiv  /-'in  i  and  slio\v 
tliat  t\vu  inveixc  points  with  respect  to  /•'  coi-iespoiid  to  two  lines  in 
>',  which  are  conjii^ate  polars  with  respect  to  the  line  complex  which 
corresponds  to  /•'. 


CHAPTER   XX 

GEOMETRY  OF  N  DIMENSIONS 

163.  Projective  space.    \Ve  shall  say  that  a  point  in  n  dimensions 
is  defined  l»v  tin'  //  ratios  of  n  -f- 1  coordinates;   nainelv, 


(V) 

The  values  of  thi'  coordinates  may  be  real  or  imaginary,  but  the 
indeterminate  ratios  0:0:--':U:0  shall  not  be  allowed.  The 
totality  of  points  thus  obtained  is  a  space  of  n  dimensions  de- 
noted bv  \. 

A  straight   line  iu  \  is  defined  liv  the  equations 

pj\  =  .//i+Xr,,  (/  =  !,  2,  ...,/<+!)  (-2*) 

where   //r   and    ^   are   constants   and   A    is   an    independent   variable. 
(  )bviously  //(  and  zi  are  coordinates  ot  two  points  on  the  line,  which 
is   thu>   uniquely  determined    bv  auv  two  points  in   ,s\    Also,  any 
two  points  of  a  straight  line  may  be  used  to  define  it. 
A  plane  in  \  is  ddined  by  the  equations 

ps—ffi+XZi  +  pit';,          (4=1,  2,  ...,  H+l")  (•}) 

where  //,,  z:,  n\  are  the  eoi'irdinates  of  three  points  not  on  the  same 
straight  line,  and  X,  fj.  arc  independent  variables.  Therefore  a  plane 
is  unii|iiclv  determined  by  anv  three  noncollinear  points  of  ,s':i,  and 
an\'  three  siieh  points  on  a  plane  mav  be  used  to  detine  it. 

In  general,  a  manifold  of  /•  dimensions  lying  in  .S'n  may  be  defined 
bv  the  equations 


X  //'.-  4-  • 


where  //  are  eonstants  not  connected  bv  linear  relations  of  the  same 
form  a--  (  1  »,  and  X,  are  /•  indcpcndelil  \ariables.  Such  a  manifold 
is  called  a  ////-.//•  .vy/i/r,  -./'/•  ill  iin-iisi'  i/is  and  \\'ill  be  denoted  bv  .S'r'. 
It  i-  alxi  railed  an  /•-//'//.  A  straight  line  is  therefore  a  linear  space 
ot  one  dimension  <  .s','  ).  a  plane  is  a  linear  >paee  of  t  \\  o  dimensions 


1'OINT  COOKDINATKS 

('•S'.,'),  and  ,S't  itself   is  a   linear  spurt-   of   //   dimensions.     From    the 
drliiiit  inn    follow  at    oiirr   tin-   theorems: 

/.  ,1   lint'iir  xjnii-i'   nt   r  Ji uii'iixinnx   ix   n  n  iifiii'l  i/  ilitiriiniiiil  I**/  iini/ 
r  -\-  1  i>nintx  at  Sn  iinf  ////iii/  />/  it  hiiiiir  ftjiiift'  nt  /n/t'i'/'  ili/m -iixin/ix,  ni/i 
(in//  /•  -f-  1    jiuiiitx  nt'  tin  ,s''  mill/  In-   iixi-il  fn  ili'tliii'  it. 

II.  .1  lint'iir  K/iHd'  nt   /•  (Itiin'itxt'ititi  ix  ili'ti'i'iniiiiil  1'if  it   Itiii'iir  .*-•/»/<•>' 
of   r  —  ]   (hnu'iiiscuix  /i/ii/  iini/  fiiitnt  nut   in  tl/iit  hitter  xjnii-i'. 

It    is  t-asv  ID  srr  that  a  linrar  spare  of  n       \  dimensions   is  also 
defined  1>\  a  linrar  etjuation 


which  is  analogous  to  the  equation  of  a  plane  in   three  dimensions. 
An   ,S';'    ,  is  therefore  called   a  7t//j>t'i'j>/<itH'. 

It  is  also  easy  to  see  that  the  coordinates  j\  which  satisfy  eqiia- 
tions  (4)  satisfy  n  —  r  equations  of  the  form  (  "> ),  and  ronverselv. 
Therefore 

///.  .1  littfitr  x[><i<;>  lit'  r  ill  iiii'itxinnx  null/  In-  il,'t;n<  <l  lij  n  —  r  liult'- 
jit'/ntt'/it  ///ifiir  i'i/i/iiti"/ix,  iini/  tx  t In  ri'J art'  tin'  i/iti'/'xi'i-t/n/i  i>t'  n  —  r 
/It/jn'rji/il/it'X. 

In  A'ri  we  shall  be  interested  in  projertiye  geometry:  that  is,  in 
properties  ot  the  space  which  are  unaltered  by  the  transformation 


where  the  determinant  <<ik  does  not  vanish.  Accordingly,  it  we 
are  concerned  with  Lj'romrtrv  in  an  A''  we  may  equate  to  -\,.,.,, 
.V.  ..,  •  •  •,  Xtt  ,  ,,  respectively,  the  left-hand  members  of  the  it  —  r 
equations  which  define  it,  w  hile  lea\"in^  ./•  .  ./•.,,  •  •  •,  ./;  ,  unchanged. 

No\\'    placing  .\'r  ....    -\'r  ;  .,,•••,   -V;i  ,  ,    equal    to    /ero.    We    have    left    the 

/•  -1-1  homogeneous  coordinates  ./'j,  ./•.,,  •  •  •,  .r(,  ,  to  drlinr  a  point  in  N' . 
It  follows  that  an  S',  is  an  X^  \\  ith  a  smaller  number  ot  dimensions, 
and  that  any  projertiye  properties  ot  .s';i  which  are  independent  o| 
t  he  value  ol  //  apply  to  an  v  X'. 

Resides    the    linear    spaces    tin-re    may    exist     in    N     other    spaces. 
Such  spaces  may  be  defined    by  equations  ot    the   torm 

p.i   =  (/>  <  Xr  \...  •  •  •,  X   ),  (  7  ) 


,°V.MJ  -V  DIMENSIONAL  (JEOMETRY 

where  (f)t  are  functions  df  /•  independent  variables  X4.  If  $_  arc 
algebraic  funetions,  equations  (7)  define  an  alijiJirnit'  *!"<>•>'.  If  we 
substitute  the  values  of  .r  from  (7)  in  the  /•  equations, 


which  define  an  N'    ,.,  \\  ~e  shall  lia\f  /•  I'ljuations  to  determine  the  /• 

\  arialilcs  \;  .  The  solutions  of  these  equations  used  in  (  7  )  ^'i\'e  the 
nuinlter  of  points  of  the  space  (7)  \\hieh  lie  in  an  >s''  ,.  Let  this 
nuiul>er  lie//.  Then  //  is  called  the  //-•<//•«•  of  tin-  space  (7),  and 
that  space  is  denoted  l»v  ^/.  \\here  /•  ,n'i\'es  the  dimensions  of  the 
space  and  //  the  number  of  [>oints  in  \vhich  it  is  cut  l>y  a  general  \'  ,.. 
Thus  N,-'  represents  a  curve  which  is  cut  1>\  anv  hvperplane  in  _</ 
points,  and  S-'t_  ,  a  hvpersurface  \\  hich  is  cut  l>v  anv  straight  line 
in  1  1  points. 

A  spaee  N'.'  mav  also  he  defined  l>v  //  —  /•  simultuneous  eijiiations. 
I  suallv  the  same  sjiace  mav  l>e  rejtresented  1>\-  eithei-  this  method 
or  I>v  that  of  equations  (7),  hut  sometimes  this  is  not  possible. 
If  N^,  is  represented  bv  a  single  algebraic  eipiation,  //  represents 
the  degree  of  the  equation.  If  N'.'  is  represented  bv  n  —  r  eipuitious, 
//  is  in  general  the  product  of  the  degrees  of  the  equations. 

In  this  chapter  \ve  shall  conline  our  attention  to  \~  ,  delined  by 
the  eoiiation 


and  sections  of  the  same. 

164.  Intersection  of  linear  spaces.  Consider  two  linear  spaces 
N,'  and  .s',.' .  A  point  ./• .  which  is  common  to  the  two.  must  satisfy 
the  'In  —  r  —  /•.,  equations  in  n -\-~\  homogeneous  variables: 


+ 


POINT  COORDINATES 

We  have  three  cases  to  distinguish: 

I.  It  '2  n  —  rt  -  /•,  >  ;/,  equations  (  1  )  have  in  general  no  solution. 
There  results  the  theorem  : 

7.  7V/;  Inii-iir  xjiiti'i'x  Sr  (inil  Sr  hiii't1  //i  iji-HiTnl  /in  jmi/it  in  I'linini'in 
ic  ft  en  i'  +  /•„  <  //. 

For  an  example  consider  two  straight  lines  in  ,s'  or  a  straight 
line  and  a  plane  in  ,s\ 

'2.  If  '1  n  —  /•  —  /•_,  =  //,  equations  (  1  )  have  in  general  one  solution. 
There  ivsiilts  the  theorem  : 

II.  T/t'n    luifiir    xjitti-i-x    S'     it/ii/    S'.     niti'i'Xi'rt    til    i/fin'/'il/   ill    "in-   ji"in> 
ti'tn  II    /'   +  /'.,  =  "• 

Kxamples  are  two  straight   lines  in  ,s'|(  a  line  and  a  plane  in  S 
and  t  wo  planes  in  ,s'  . 

:!.  It'  'In  —  /',  —  ''.,  <  "•  equations  (  1  )  ha\'c  in  general  an  infinite 
number  of  solutions.  Let  us  suppose  that  /•  +  /•„  =  /i  -f-  </.  The 
number  of  eijuations  (  1  )  is  then  n  —  it,  and  tluiy  therefore  define 
an  S't.  Thei'e  results  the  theorem: 

///.  TII'"  Inn-ill'  XjuK-i'X  Sr  /l/>il  N',  U'luTe  /"j  +  /'r  =/<  +  (/,  ilttt'l'iSt'Ct 
III  if'  lli'l'ill  in  tin  N(. 

Examples  of  this  theorem  ai'e  that  in  S:  two  planes  intersect  in 
a  straight  line,  and  that  in  >>'(  two  hyperplanes  intersect  in  a  plane. 

<  )f  course  an\"  two  linear  spaces  mav  so  lie  as  to  intersect  in 
more  points  than  tin-  above  general  theorems  call  for.  Let  us  snp- 
po>e  then  that  S'  and  ,V'  intersect  in  an  N[.  Now  >'.'  is  defined  bv 
/',  -i-  1  points,  of  which  <i  +\  ma\'  be  taken  in  S't.  Sinnlarh'.  S'r  is 
defined  b\-  /•„+!  points,  of  which  <i  +  1  ma\  be  taken  in  S't.  If, 
fore,  we  take  //  +  1  points  in  \('.  J\—  it  other  points  in  N'  but 
ot  in  .s'|.  and  /•.,  -  <i  points  in  S'_  but  not  in  .s',,',  we  have  /•,+  /'.,-//-(-  1 
points,  \\hii-h  ma\'  be  used  to  define  an  N'  u.  This  S'  ,  con 
tains  all  of  \'  and  all  of  N'  since  it  contains  /',  +  1  points  of  the 
former  and  /*.,+  !  jioints  of  the  latter. 

Thereli  ire  \\  i  •  ha\'e   the   theorem  : 

IV.    If  N'    ////-/  ,S':'    i/ttf/'iti-rt   i,i   mi    S't.  th.'il  In-   in   <//<    S'  ,        a. 

An  example  of  this  theorem  i>  that  in  N  if  t\\n  sti'ai^ht  lines 
(  .s','  )  intersect  in  a  point  (  N'  ),  thc\  lie  in  a  plane  (>'.').  Another 


ter 
n 


;>!»:2  .V- DIMENSIONAL  (iEOMKTUY 

example    is   that    in   \   if  two   planes   (  .s'_!  )   intersect    in   a  straight 
liiu'  (  N,' ),  they  lit-  in   an  .s'[. 

Conversely,  we  have  tin.*  theorem: 

V.    If  Sr   ini'l  .s'r'  lit'  in  an  X'm(tn  <  /?  )•.  tin-//  i/if>'w<-f  in  on  S'  . 

' :  i  '  i  *  '  i      m 

if  r,  -f-  r.,    i  in. 

This  is  (inly  a  restatement  of  theorem  III,  since  l>v  the  previous 
seetion  we  have  only  to  consider  the  \'_  in  which  the  t\vo  linear 
spaces  lie. 

Similar   theorems    may    be    proved    for    the    intersect  ions    of    the 

curved  spaces  N''1  and  A':'-'.    These  we  leave  for  the  student. 

'i  . 

EXERCISES 

1.  Show  that  the  hvjierjilitiics  in  ,s';i  inav  be  considered  as  points  in  a 
space  of  n  dimensions  ^/(. 

2.  Show    that    if    >',';i    contains    //  4-  1    points    of    N;'    it    contains    all 
points  of  V. 

3.  Show    that    through   any    S[.   may   lie    passed   x""*"1   liy pt-rplaiies, 
anv  /(  —  1;  <>f  which  deterniine  >\' :  that  is.  in  the  notation  of  Kx.  1  any 
S[  is  coiimion  to  a  — ,', _A. .  ]. 

4.  Shnw    that    two   algebraic   spaces    S''t  and    S1^'.   do   not    in    general 
intersect    if    ///  -)-  ///'<  /t,  and    intersect   in   an    .\'y    if   ///  -)-  /// '  =  n  -\-  n. 

5.  Show  that  every  >';,  is  contained  in  an  >','„.]. 

6.  Show  that  every  curve  of  order  y  is  contained  in  a  linear  space  of 
a  number  of  dimensions  not  superior  to  </. 

165.  The  quadratic  hypersurface.    The  equation 


detines   iu  i    .^"    ,,   \vhieli    \ve    shall    call    a   ifUdilnitii1   liifperxurfiii*?   or, 

inure    concisely,    a    >jt(i.(<(r«'.     For   convenience   we    shall    denote    the 
surface  by  c^>. 


to   values   of  A.  e^iveii    by   the 
XJV'/    ::•    =  0.  :} 


POINT  COORDINATES  :)03 

If  ^<(ij/l2li.=  ni  tin.-  points  //i  and  zt  are  harmonic  conjugates  with 
respect  to  the  points  in  which  t  he  line  (  '1  )  intersects  c/>,  and  are  called 
i'<tnjnt/iif,'  im'nitx.  Therefore,  it  //  is  lixed,  any  point  on  the  loens 


is  a  harmonic  conjugate  of  //,.  This  loeus  is  a  hyperplane  called 
the  i'1'litr  l////»T/>/<i/tt'  of  //i  with  respret  to  the  qnadric. 

If  ii  (  is  also  on  the  qmulric,  both  roots  ot  (  ;>  )  are  y.ero,  and  the 
line  (-)  touches  the  hvpersurface  in  two  coincident  points  at  //,, 
or  lies  entirely  on  <fi.  The  polar  (4)  then  becomes  the  tangent 
hvperplane,  the  loeus  of  all  lines  tangent  to  <£>  at  i/t.  In  no  other 
east.'  does  the  polar  contain  the  point  //,. 

It  follows  directlv,  either  from  the  harmonic  propertv  or  from 
equation  (  -I  ),  that  it  a  point  /'  is  on  the  polar  of  a  point  (,',  then 
(t>  is  on  the  polar  of  /'. 

More  generally,   let  //,  describe  an   S'r  defined  bv 

P.'/i  =  U\  ''  '  +  ^i//."  '  +  •  •  •  +  X,.//'/'  '  ".  (  5  ) 

The  polar  hvperplanes  are 


\  allies  of  .rr  co  in  n  ion  to  these  hvperplanes  sat  is  fv  the  /•  +  !  eipi  at  ions 
^",///.'V-n<  (/-    1,  'J,  •  •  -,  r  +  1  )  (»J) 

and  therefore  form  an  /\'_,._,.  The  t\\o  sjiaces  ,S';'  and  .V'  r_,  are 
I'unj  ii'/nti'  /inlnr  xiiitcfi*.  Kach  point  ol  one  is  conjugate  to  each 
point  of  the  other,  ('on  jugate  polar  lines  in  A'.,  form  a  simple 
exam]  ile. 

If  the  equation  of  the  polar  hvperplane  is  written  in  the  form 


pitk  -        ",A//,-  (     ) 

i 

I.ct  us  consider  first  the  case  in  which  the  determinant  <i  t  , 
\\hich  is  the  tl/vi-ri//iiii<oit  of  (  1  ),  does  not  vanish.  Then  if  the 
(plant  it  it's  it,  in  (7)  are  replaced  bv  /.ero,  the  equations  ha\e  no 
solution.  1  heretoi'e  all  possible  \alues  ot  //  >_;'i\e  definite  values  ot 
/',  which  cannot  all  become  /.ero.  A^'aiu,  equations  (7),  as  they 


-V  DIMENSIONAL  (\EOMETRY 

stand,  can  IK-  solved  for  //,,  so  that  any  assumed  values  of  n(.  deter- 
niiiu'  unique  values  of  //,  which  cannot  all  lie  y.ero.  Summing  up, 
\vi-  have  the  theorem  : 

//'  tli,'  ilixrrhnhmnt  <>f  </>  rA^-x  n»t  vanish,  erf/-//  [mint  »f  Sn  Ixm  n 
dt'finiti'  i>"l'ir  lti/]>t'rj>l<tnt',  <tnil  ct't'ri]  hyperplane  in  N(  /x  (lie  /><>/<>/• 
ct'  (i  di  finite  jxit/tt.  In  piirticuhtr,  itt  t'Ct'fif  pmnt  »J  0  tlici'i'  /x  <t 
d>  fluid'  tititi/fiit  /i/itiii'. 

Consider  now  the  case  in  which  the  discriminant  u/,!  vanishes. 

I       'A.  I 

There  will  then   he  solutions  of  the  equations 

2"*&=0.         (A-  =  l,  I',...,  //-f  1) 
.   i 

Any  point  whose  coordinates  satisfy  (S)  lies  on  $,  since  its 
coordinates  satisfy  the  equation 


and  is  called  a  singular  point  of  $. 

(  )l)viously,  at  a  singular  point  the  tangent  hyperplane  is  indeter- 
minate, and  in  a  sense  any  hyperplane  through  a  singular  point 
may  he  called  a  tangent  hyperplane. 

Equation  (  •>  )  shows  that  any  line  through  a  singular  point  cuts 
the  qliadric  in  two  points  coincident  with  the  singular  point,  which 
is  thus  a  double  point  of  the  quadric.  It  also  appears  from  (  -I  ) 
that  any  point  of  (f)  may  be  joined  to  any  singular  point  by  a  straight 
line  lying  entirely  on  <£>. 

Any  point  //,  not  a  singular  point  has  a  definite  polar  hyperplane 

X-       n    •    \     f    ,       n    •   I 

V    •;     V"  ,'/    -./•,  =  0  ; 

^^^  ^—        ' 

*    1  ,1 

and  since  this  may  be  written 

i    -     n    •    I     (   *•        „    f   1  ] 

V  j   V  ,/,,,-.   //,    o, 

,   1    I.  *•  -  1        J 

it   passes  through  all  the  singular  points. 

The  number  of  the  singular  points  of  (f)  will  depend  upon  the 
vanish  ing,  or  not,  ot  the  minors  of  <i  tk  .  In  the  simplest  case,  in  which 
n  ^  vanishes  but  not  all  of  its  lirst  minors  vani>h,  ecjuations  (S) 


POINT  COORDINATES 

have  one  and  only  one  solution,  and  (/>  has  one  singular  point. 
Therefore  the  quadric  consists  of  x "  '  lines  passing  through  the 
singular  point. 

Suppose,  more  generally,  the  minors  of  n it  which  contain  //  +  2  —  r 
or  more  rows  vanish,  but  that  at  least  one  minor  with  u  +\  —  r  rows 
does  not  vanish.  The  equations  (S)  then  contain  /i  —  r  +  \  inde- 
pendent ('(illations,  and  the  singular  points  therefore  form  an  X'r_r 
The  quadric  is  then  said  to  be  r-f<>t'l  x]»'<'inliz<'(J.  The  number  r  is 
so  chosen  that  a  onefold  specialized  quadric  has  a  single  singular 
point,  a  twofold  specialized  quadric  has  a  line  of  singular  points, 
and  so  on. 

Any  S'r  which  is  determined  by  the  X'r  _j  of  singular  points  and 
another  point  /'  on  (f>  lies  entirely  on  (/>.  This  follows  from  the  fact 
that  all  points  of  the  ,S';'  lie  on  some  line  through  /'  and  a  singular 
point,  and,  as  we  have  seen,  these  lines  lie  entirely  on  <£.  In  par- 
ticular, if  r  =  2,  the  quadric  consists  of  planes  through  a  singular 
line  ;  if  r  =  •  >,  the  quadric  consists  of  spaces  of  three  dimensions 
through  a  singular  plane:  and  so  forth. 

A  group  of  n +~[  points  which  are  two  by  two  conjugate  with 
respect  to  c£  form  a  self-conjugate  ( //  +1  )-gon.  There  always  exist 
such  ( // -f-1  )-gons  if  the  quadric,  is  nonspeciali/ed.  This  may  be 
seen  by  extending  the  procedure  used  in  Jj  1*2.  By  a  change  of 
coordinates  the  it  + 1  hyperplanes  which  are  determined  by  each 
set  of  /^-points  in  the  (//-f-l)-gon  may  be  used  in  place  of  the 
original  hyperplanes  j\  =  <X  In  the  new  coordinates  any  point 
whose  coordinates  are  of  the  form  ./•i(.  =  l,  .>•,=  ()  (i~---k)  has  the 
hyperplane  rA.  =  0  for  its  polar.  The  equation  of  <f)  then  becomes 

Now  the  vanishing  of  the  discriminant  and  its  minors  denotes 
geometric  properties  which  are  independent  of  the  coordinates  used. 
Hence  we  infer  that  for  the  general  quadrie  all  the  coefficients  /-, 
differ  from  y.ero.  If  the  quadric  is  /--fold  specialized,  it  may  be 
shown  that  equation  (I*)  may  still  be  obtained,  but  that  /'  of 
the  coefficients  vanish. 

It  the  quadric  is  general,  by  another  change  of  coordinates 
equation  (I*)  may  be  put  in  the  form 


.V   DIMKNSIONAL  CKOMKTRY 

EXERCISES 

1.  I'rove  that  all  points  of  any  >','  through  the  N'  ,  of  singular  points 
have  the  same  polar  hvperplane,  whieh  passes  through  >''.  ,.  and  that, 
conversely,  any  hyperplane  through  the  singular  S'r  ,  lias  for  its  pole 
anv  point  ot  a  certain  S'r. 

'2.  Show  that  for  any  ipiadric  which  is  r-fold  speciali/ed.  anv  tangent 
h\  "pei-plane  at  an  ordinary  point  is  tangent  to  the  ipiadric  at  all  points 
of  an  >','.  lying  on  <£  and  determined  by  the  point  of  contact  and  the 
singular  >','.  _  ,. 

3.  Show  that  if  <$>  is  more  than  once  specialized,  any  hvperplane  is  a 
tangent  hvperplane  at  one  or  more  of  the  points  of  the  singular  >','.    ,. 

4.  Prove  that  every  >','„  through  a  point  //.intersects  <£  in  an  >'„;'  ,  and 
intersects  the  polar  hvperplane  of  ?/•  in  an  >','„  _1?  which  is  the  polar  hvper- 
]ilane  of  ;/,  with  respect  to  the  ^',,^_}  in  the  space  S'>n. 

5.  Prove  that  if  >',' and  V  _  r    \  are  conjugate  polar  spaces,  the  tangent 
hvperplaiies  to  <£  at   points  ot   the   intersections  of  <£  with  one  of  these 
are  exactly  the  tangent   hvperplanes  of  <£  which  pass  through  the  other. 

6.  Trove  that  any   plane   through   the   vertex   of  a   hvpereone   inter- 
sects  it    in  general   in  two  straight  lines,  but   that    if  /;  ~^  '<\  it  mav  lie 
cut irely  on  the  hvpereone. 

166.  Intersection  of  a  quadric  by  hyperplanes.  Let  (f>  be  a  (piadric 
hvpersurface  in  ;;-space  with  the  equation 

2"'*"'v*=°-       K,-=ff*)  d  > 

It  is  intersected  bv  any  hyperplane  //  in  a  (piadric  hypersurfaco  <f>' 
lvin<4  in  //.  To  prove  this  we  have  simply  to  note  that  the  equation 
of  //  may  be  taken  as  •>'„.]="  without  changing  the  form  of  (1  ). 

\Ve  proceed  to  determine  the  conditions  under  which  (/>'  is  spe- 
cialized. If  (}>'  has  a  singular  point  /',  any  line  in  //  through  /' 
intersects  $>' ,  and  therefore  (/>,  in  two  coincident  points  in  /'.  There- 
fore, either  //  is  tangent  to  <£  at  /',  or  /'  is  a  singular  point  of  (/>. 
( '(inversely,  if  //  is  tangent  to  0  at  a  point  /'.  or  it  //  passes  through 
a  singular  point  /'  of  (f),  then  <$>'  has  a  singular  point  at  /'. 

If  (/>  is  a  noiispeciali/.ed  (piadric.  the  hyperplane  //  has  at  most 
one  point  i  if  tangvncv.  I  Iciice  : 

7.  .1  ii'inxfii'i'liiUzi'il  ijiiiiil ri<'  is  iiiti'i'xi'i'tt'il  f>'/  "/>//  //"///'//////•///  /////«/•- 

jiliiii'  in  <(  ii'iiixjii 'i-nili'i  '/  nlttliJrii'  "/  "//*'  1'iii'iT  </t /ili'timutl,  till'!  /# 
intiTX'-ffi-il  l>i/  'i  (<///</>  >/t  /////'«  /•/'I'l/ii1  i/i  'i  inii-i'-x/n'i-inhzi'il  tjuiii/rif 
H'itJi  it*  siiti/iil'ir  fi'tnt.  "f  lln'  i>"int  <>t'  til ii'!>'n<-ij. 


POINT  COORDINATES  --J'.IT 

If  the  (ptadrie  $  is  once  speciali/.ed,  having  a  singular  |i<>int  A, 
any  hyperplane  which  is  tangent  to  </>  at  a  point  />'  distinct  from  ./ 
is  also  tangent  to  c/>  at  all  points  of  the  line  A  /•'  (  Kx.  _,  £  l*i  •">_). 
Hence  : 

77.  If  t/ii'  <fi<<i</rft'  (/>  /MX  "til-  xhi'/ii/i/r  finhif  .1,  '/////  hyperplitn?  wlii<-h 
tjufx  n»f  pax*  tfirmnjh  A  infi-rxt'i-fx  (p  in  <i  nnnftpt'i-iii/izi-it  ^iciilrli'  <>f  ,,/t,- 
InH'tT  (Unu'nxinii  ;  tint/  liyprrplrine  throiujh  A  />ut  n<>t  t'ltt^i'nt  <tt  <nn/ 
dt/n'r  point  infi-rtH'rf*  $  in  <i  nncc-speciuUzt'd  y?w//vV,  with  n  sini/ttl'ir 
pnint  nt  A;  <in<l  <ni>f  hyperplnn?-  t<ni>i»'tit  <tl»n<i  tin'  liin1  A/!  int>'rxi''-f* 
0  in  a  ftrirc-sju'i'iii/izt'i/  <p«i<~lri<!  in'f/i  tin-  line  A  /•'  «*  <(  suii/idar  //'/!>•. 

More  generally,  let  $  he  an  r-tdM  specialized  quadric  containing 

a  singular  >','_,,  \yhieh  \\'e  shall  call  S.  Any  hyperplane  meets  S  in 
an  S'r  .,  or  else  completely  contains  S.  Mnreoyt-r,  if  //  is  tangent  to 
<f)  at  some  point  /'  not  in  A',  it  is  tangent  at  all  points  of  the  A' 
determined  hy  /'and  S,  and  therefore  contains  S.  I-'roni  these  faets 
\ye  haye  the  following  theorem: 

777.  If  t/n1  tji/riiln't'  (f)  /x  r-fniil  ^^i-cin/i^n/,  fmrhif/  n  xinr/uldr 
(  )'  —  1  )-tl<tt  X.  untf  hyperplane  //  nut  cnntnitiini'f  S  hiti-n«'<-tx  <f)  in  <n> 
(r  —  \)-fi>hl  npfvwUzcd  i/u<n/r/'/'  it'h<>x<'  »in;fitJnr  (r  —  "l)-tltt  ix  th>' 
inti'wrffnn  <>f  II  ,tn<l  S:  nni/  hifprrplnni1  <-")it<iinin<j  Slut  n»f  fii/i;/<'nt 
t"  (£>  infcrsi'rfx  (f)  in  mi  r-t'<>/</  ttpi'i'iftUzwl  ijn</i/ri<'  //7/">v  xim/ultir 
(  r  —  1  )-fi<tt  i*  S;  mill  nni/  Itifpcrplftn 
iin  (r  +  1  )-f»l<l  spccinlizcil  ijumlrin  u' 
Lit  I'  (ti)-l  S. 

Consider,  now,  the  intersection  of  ^>  and  the  two  hyperplanos 

V,  ,,,•.=  (),       2v,-=o,  (•_>) 

\\'liich  \ye  shall  call  //  and  7/o  respectively.  //  intersects  0  in  a 
rjuadric  c/>'  lying  in  Sn  p  and  //.,  intersects  </>'  in  a  (piadric  0".  which 
lies  in  the  •*>'„_.,  formed  hy  the  intersection  of  //,  and  //.,.  Hence 
the  common  intersection  of  the  ipiadric  (  1  )  and  the  hvperplanes  (  '_'  ) 
is  a  tpiadric  of  /)  —  :>>  dimensions  lyin^-  in  a  space  of  //  —  '1  dimensions. 
'1  his  ijuadric  is  also  the  intersection  of  the  (piadric  determined  hy 
(f)  and  //]  and  that  determined  hy  c/>  and  //„. 

This  ipiadrie  may  also  he  ohtained  as  the  intersection  of  (/>  and 
any  t\yo  hverlanes  of  the  encil 


V(//  +\/.,)r  =0, 


.V  DIMENSIONAL  (JKOMKTKY 

in  which  thnv  ;uv  in  general  two  hyperplancs  tangent  to  $  and 
fixing  two  points  of  taii;j;ency  on  <p.  Hence  we  have  the  theorem: 

IV.  Tin-  intfrxt't'tinn  <>f  <i  <]int'/ri<'  xnrt'ii<-i'  <£>  /,//  ,ui  ,V'  „  f  urinal  />// 
tii'n  hi/in'r}>l(iHt'x  1-nnxisfs  In  i/i'tii-ni/  >>t'  uti  .V"' ,  f urinal  lij  tin-  intcr- 
aa'fioii  i>t  t/r<>  Jiii jn'ri'oin'x  liftntf  <>n  </>.  Thi'  >\',~'.t  Jinx  flic  j>ri>j>t't'f// 
t/nit  'i/i//  ji"hit  i>n  if  >/ni//  nt'  jn'nii'il  tn  I'di-h  lit'  ttru  fi.i'fit  [mi/ifx  nn  (p 
1'if  *fr<ti<//it  li/tt1*  li/ini/  entire///  an  (f). 

Of  course  the  fixed  points  and  the  straight  lines  mentioned  do 
not  in  general  belong  to  the  X'"' ... 

We  shall  examine  this  configuration  more  in  detail  for  the  ease 
in  which  0  is  not  specialized,  and  shall  assume  the  equation  of  (f) 
in  the  form  ^  ., 

2/r^0-  (1> 

Then     the    condition    that     a    hvperplane    of    the    pencil    (o)    is 

tangent   is  ^    ,  .^  ,.^, 

2"?  +  -  ^^  "/',  4-  X  N  f'j  =  0.  ( .» ) 

If  the  roots  of  equation  (  ">  )  art1  distinct,  there  arc  two  tangent 
hyperplanes  in  the  pencil  (  •>  ).  and  we  have  t  he  general  case  d esc n lied 
in  theorem  I\.  It  the  roots  ot  (  .>  )  are  e([Ual,  there  is  only  one 
tangent  hvperplane.  and  the  corresponding  livporcoiie  on  <f>  is  not 
snt'ticient  to  determine  the  •s','1"_!:i,  hut  must  he  taken  \\ith  another 
h\'pcrplane  sect  ion. 

Finallv,  equation  (•>)  mav  lie  identically  satisfied.  This  happens 
when  - -i  ,  .  ^  - .-,  , , 

y lt-  =  o,     V,,  /,  =  o,     v//-  =  o,  ( i;  > 

*4     '  +- ~i     '    '  —,' 

which  express  the  facts  that  each  of  the  hyperplanes  //  and  //, 
j^ivcii  Kv  ('(juations  ( '_' >  are  tangent  to  0.  and  that  the  point  of 
tan^encv  of  each  lies  on  the  other.  Then  anv  one  i>f  the  hvper- 
plancs  nt  the  pencil  (  •)  )  is  tangent  to  (/>.  and  the  jioint  ot  taiM^ciicv 
is  "_  -4-  \/(,.  so  that  the  points  ot  taii'4'encv  he  on  a  straight  line. 
The  pencil  of  hyperplanes  (  '•}  )  consists,  therefore,  of  the  liyperplanes 
taiiLTent  to  (f)  at  the  points  of  a  straight  line  on  (/).  Let  us  call  this 
line  h.  Then  all  points  on  the  X;  .,  determincil  liv  (/>.  //,.  and  //., 
mav  lie  joined  to  ;mv  point  of  It  liv  means  of  a  straight  line  IviiiLj1 
on  (/).  Let  ;i .  lie  a  point  on  X  J  „.  Then  anv  point  on  the  line  joining 
//  to  a.  point  ot  //  is  it  +  \A  4-^t//.  The  coJ'irdinates  of  this  point 
sat  isfy  ei piat  ions  (  '2  )  and  (  I  )  l»v  \  ill  ne  of  (  ('>  )  and  t  he  hypothesis 


POINT  COORDINATES  :-)!)«.) 

that  v/  satisfies  these  equations.    Consequently  in  this  case  S:i2   ,  is 
a  speciali/ed   qiiadric  with  //  as  a  singular  line. 

('(insider,  now,   the  intersection  of  (/>   by  an    .s';'    ,   defined  bv   the 

hyperplanes         NV.,-:=0,      V/,  ,•-:(),      Vr./-^  0.  (7) 

These  detennine  witli  c/>  an  •s';,"'4.  which  niav  also  be  determined 
as  the  intersection  <if  </>,  and  anv  three  linearlv  independent  hvper- 
plaiics  of  the  bundle  delined  by 


Ainoncr  these  there  are  -f.  '  tangent  hvperplanes.  If  the  ei|uation 
of  </>  is  in  the  form  (  I  ),  the  tangent  hyperplanes  are  !_n\eii  bv  values 
of  X  and  ft,  which  satisfy  the  equation 


and  the  points  of  tanp'iicy  of  these  hyperplanes  are  then  at- 
These  points  of  tan^ciicy  therefore  form  an  .s'j"1,  or  curve  of  second 
order  Ivin^  on  (/>,  and  every  point  of  the  >s',,"^4  which  we  are  con- 
sidering mav  be  joined  to  each  point  of  this  curve  by  a  straight 
line  on  (f). 

Filiation  (  *  )  is  identically  satisfied  when  each  of  the  hyper- 
planes (7)  is  tangent  to  (/>  and  the  points  of  tan^eiicv  of  each  lies 
on  the  other  two.  Kadi  hvperplane  (X)  is  then  a  tangent  hvper- 
planc,  and  the  points  of  tan^encv  are  <t t  -\-  X//_  -)-  /JLI\.  where  X,  //  are 
unrestricted.  The  bundle  therefore  consists  of  all  hvperplanes 
whose  points  of  tan^vncv  are  the  points  ot  a  plane  Ivin^  on  cf>. 
Therefore  each  point  ot  the  St~_ '4  is  joined  to  each  point  of  this 
special  plane  bv  lines  Iving  on  c/>  and  on  the  St~ 4.  Therefore  the 
Sn2  (  is  in  this  case  a  specialized  qiiadric  with  that  plane  as  a 
singular  plane. 

('onsider,  now.  the  general  ease  ot  the  intersection  of  ^>  b\-  the 
S'ti  _k  defined  b\  the  /•  hvperphuies 

2".'-'-,       °-  (/-  1,  I',  ..-./')  (10) 

This  is  an  >',;  /  ,.  which  mav  also  be  obtained  as  the  intersection 
ot  c/>  and  anv  /•  hvju'l'planes  ol  the  s\stem 

in  \\lneli   there  arc  "vnerallv    /"     "'  taii"'ent    hvperplanes. 


4(10  .V  DIMENSIONAL  UKOMKTKY 

In  fact,  if  we  limit  ourselves  to  a  nonspecialized  0  and  take  its 
equation  as  (  t  ).  the  condition  that  a  hyperplane  (11)  should  be 

tangent    is  ^ 

N  (,,;i.+  Xl,^'+...  +  \<  _,„;'•' r=rt,  (12) 

and  the  points  of  tangeiicy  are  then  '/'/'-f-  A,^-'-f  •  •  •  4-  A,.  ,'/'/'', 
where,  of  course.  A,  satisfy  (  1  '2  ).  These  points  form,  therefore,  a 
S.~ ' .,  on  (f>.  and  anv  hypereone  with  its  vertex  on  this  N^.,  passes 
through  the  N'J  ,  ,  which  we  are  discussing.  \Ve  have,  therefore, 
t  he  theorem  : 

V.  Tli'  / /ifi'/'Xf'i'f  t"it  "t  it  n<inxp>'<'tnhzc<1  fiiinilt'ii'  <$>  f>i/  'in  S  {.  <l>'fi)ii  <1 
t>ii  /"  hin)<'ri>l<nit'n  /*  '///  ^,~\.  ,  n'/m'Ji,  in  </i'//i'/-ii/,  //i/x  t/ir  i>r"jn'rti/  t/i/tf 
,//••//  i>t'  if  ft  i><>nitx  miiji  In'  j"iiii-<l  fn  1'Hi'li  jKiint  "fa  wrhihi  X'^.,  "»  (f) 
/•//  xtrniijltt  Inn'*  ////////  »n  0. 

According  to  this  tlieorem  we  have  on  (ft  two  spaces.  ,V-_| ^ ._ l  and 
,\J  .,.  such  that  each  point  of  either  is  connected  to  each  point  of 
the  other  bv  straight  lines  on  0.  It  is  obvious  that  the  condition 
must  hold  ~2  :  /,  -  n  —  1. 

If  a --'-\.  the  t\\o  spaces  are  ,v,:  and  ,S',J',  each  of  which  con- 
sists of  a  pair  of  points.  It  n  =  4.  the  two  spaces  are  ,s',jl  and 
,s '-  .  one  of  which  is  a  curve  of  second  order  and  the  other  a  pair 
of  points.  If  n --  ~).  we  have  either  an  .s'/'  connected  by  straight 
lines  witli  an  N;'.  or  an  ,s',Jl  connected  in  a  similar  manner  with 
another  .s',-'. 

In  the  first  and  last  of  the  examples  just  given  we  have  two 
spaces  of  the  same  number  of  dimensions  occupying  with  respect 
to  each  other  the  special  relation  described  in  the  theorem.  In 
order  that  this  should  happeii.il  is  necessary  that  t>  —  k  —  1  —  /-  —  -  : 

»  +  1 
whence   /•  =  •      Hence   it    is   onlv   in   spaces  of    odd  dimensions 

that  two  quadric  spaces  ot  an  equal  numbei1  of  dimensions  should 
so  lie  on  the  quadric  &  that  cadi  point  of  one  is  connected  with  each 
point  of  the  other  bv  straight  lines  on  c6.  The  number  of  dimen- 
sions of  these  spaces  is  one  less  than  half  the  number  of  dimensions 
of  the  quadric. 

Returning  to  equation  ( 1  "J )  we  see  that  it  is  identically  satisfied 
when  the  hvperplaiies  (l(i)  are  each  tangent  to  <+>  and  the  point 

of  taHLTellcV  of  each   lies  oil   e;ich  of  the  ot  llel'S.      TllCll   the  system  (11) 


POINT  rouKWNATKs  -lui 

consists  of  hyperplanes  tangent  to  (/>  at  the  points  of  an  N,'  Kiti'j; 
on  <f>.  The  ,V'^t  ,  determined  by  0  and  (  1<>)  is  then  a  /--!'«>]<  1  >pe- 
eiali/ed  (juadrie  with  the  aforcincntioiicd  N^'  ,  as  a  singular  locus. 
167.  Linear  spaces  on  a  quadric.  It  is  a  familiar  fact  that  straight 
lines  lie  on  a  quadrir  in  three  dimensions.  \\  e  shall  m-i  ieralr/.e  this 
property  by  determining  the  linear  sjiaees  which  lie  on  a  qtiadrie 
in  »  dimensions.  Let  the  quadric  $>  lie  j^iveii  as  in  x  1'i'i,  and  let 
S'r  be  a  linear  space  detined  by  the  n  -f  1  equations 

p.rt  .=  //','   +\i/'f'+  •  •  •  +  \..'/~,'  '"•  (  1  ) 

The   necessary  and   sufficient  condition  that   r,  of  (1  )  should   lie 
on  (p  for  all  yahies  of  \(.  is  that  //,  should  satisfy  the  /•  +  !  equations 

"  ..,  /•  +  ]  )  (-2) 


, 
and  the  ecpiations 


of  \yhieh  the  first  set  express  the  fad  that  each  point  //,  is  on  $. 
and  the  second  set  say  that  each  point  is  in  the  tangent  hyper- 
plane  to  (^  at  each  of  the  other  points. 

Take  any  point  /,'  on  <f>  and  let  '/',  he  the  tangent  liyperplane 
at  /p  Then  7\  intersects  (f)  in  a  specialized  qnadrie  N^1  ,.  Take  /'. 
any  point  on  .V,'^.,.  The  line  /,'/„'  tlien  lies  on  ^  liy  the  conditions  (  '1  ) 
and  (  '•}  )  and  on  ,V(-'  .,.  heeanse  \"  .,  is  speciali/eil.  The  hyperplane 
7'.,  taii'jvnt  to  0  at  I'  is  also  tangent  to  N,',J_\,  and  intersects  the 
latter  in  an  >'„";.,  \yhieh  contains  /,'/.'.  7'.,  \\'ill  also  euntain  other 
points  of  .s';i"  3  if  »  —  '•}  "^  1  :  that  is,  //  ,  •  4.  If  t  his  eondit  i<  >n  is  met, 
take  /'  in  N^1,,  hut  not  in  /,'/.'.  The  three  points  /,'.  /.,'.  /'  determine 
an  N',  which  lies  on  c/)  l>y  \'irtne  of  equations  (  'J  )  and  <  ;1  ). 

The  h\pe!'plane  7'.,  which  is  tangent  to  0  at  /'..  is  al>o  tan- 
gent to  -V..;  and  intersects  it  in  an  .V"'4  which  contains  \.'.  It  will 
contain  other  points  of  N,,J>.,  if  "  —  1-  .,-•-:  that  is.  //  •  (i.  If  this 
condition  is  met  we  may  take  another  point.  /,'.  on  this  N  (  hiii 
not  on  .s'.!.  The  four  jioints  /',  /'.  /'.  /,  no\\-  determine  an  N'  \\hich 
is  on  <f)  hy  the  conditions  (  il  )  and  (  '•*>  ). 

This  process  may  be  continued  a-.  loii;_r  as  the  condition  fur  the 
yalne  ot  //  found  at  each  step  is  met.  Suppose  we  have  determined 


402  -V   DIMKNSIONAL   < ',  K<  >.M  KTKY 

in  this  wav  an  S'r_  ,  lying  on  c/>  by  means  of  r  points,  the  tangent 
hyperplanes  at  which  have  in  common  with  (f>  an  N',^,.  ,  contain- 
ing \'  r  If  w  —  r  —  1  >/•  —  !,  that  is,  if  r<  ,  this  ,V-\.  t  has 

points  which  arc  not  on  >>','_!•  Take  /',  ,,  one  such  point.  It  deter- 
mines with  >>','._!  an  N;'  lying  on  ^>.  The  process  may  be  continued  as 

long  as  /•  <      .  but  not  longer.    Since  the  dimensions  of  the  quadrie  (/> 

are  //  —  1,  we  shall  write  the  condition  for  /•  as  r~-  -  and  state 
the  theorem  : 

/.  ,1  ntnixix't'inlizfil  t/undnt'  fontum*  linenr  .vy/'v.v  <>f  <nut  number 
nf  (linu')ttii<>ni<  ciiunl  t»  <>r  If**  tlmn  nn/f  tin'  >/ii/nf><'r  nf  dinii'itiiintix  i>t 
tin'  iiuaJrii',  t>ut  I'nnttiniH  n»  lint'rti'  sfxict'  <>t  r/r<'<ifi'r  lUwi'nsions, 

To  find  how  many  such  linear  spaces  lie  on  the  quadric,  we  notice 
that  the  point  1[  may  be  determined  in  x"  '  ways,  the  point  /_!  in 
x"~~  ways,  and  so  on  until  finally  the  point  /' . ,  is  determined 
in  x"  '  ways.  The  r  + 1  points  nun"  therefore  be  chosen  in 

x   -  "  ways:  but  since  in  any  S'r.  /•  +  ]  points  may  be  chosen  in 

x''"'  +  1'  ways,  the  total  number  of  N'  on  the  qnadric  is  x  - 

The  number  of  .\'  which  pass  through  a  fixed  point  may  be 
determined  by  noticing  that  with  /,'  Iixed,  the  /•  points  /.',  .  .  .,  /'4l 

may  bi>  determined   in  x "  ways,    and   that    in   any   ,V'.  the   r 

points  may  be  chosen  in  -x'"  ways,  so  that  the  number  of  different 

,s'^  through  a  point  is  x."  .     \Ve  sum  up  in  the  theorem: 

H'Jiii'fi  x  "  JKIXX  fh>'<>ni/Ji  <i ni/  ti.i't'iJ  jinint  <>)i  f/H'  <t iini? n<'. 

If    >i    is    odd,    the    greatest    value    of    /•    is  .    and    there    are 

x4'       '  linear  spaces  of  these  dimensions  on  the  quadrie:   it    n  is 


// 


/ 
even,  tin-  greatest   value  of  r  is  .  and  there  arc  x  '         Jl  l 


near 


spaces  of  these  dimensions  on  the  quadric. 

Let  us  consider  more  in  detail  the  case  in  which  //  is  odd.  and 
let  us  place  //--'_'/'  +  1  .  \Ve  shall  limit  ourselves  to  a  non- 
speciali/.ed  (|iiadrie  <f>  and  shall  write  its  equation  in  the  form 

«><        ...       »/        -.'-.    ------  ,-       =  0  4 


1'oINT  HM"> KDINATKS  40:; 

us  mav  be  done  without  loss  of  <_reneralit  v.  The  linear  space  of 
the  largest  number  ot  dimensions  on  (/>  is  then  N',  and  its  equations 
ma  he  written 


where   the   coefficients  satisfv  the   relations 


In  fact,  any  ,s''  is  delineol  hy  /'  +  1  linear  e(|iiations  connecting  the 
vuriuhlt'S  i>/;  ami  ./•  .  and  these  (-([nations  may  he  put  in  the  form  (  •">  ), 
provided  no  one  of  the  variahles  u  t  is  inissiug  from  the  equations. 

But  if  one  of  these  variables  is  missing,  it  is  clear  that  the  A''  cannot 

•  /' 

lie   on   (;">).     The   conditions  (  •>  )   are    found    hy  direct    substitution 
from  (  .»  )  in  (  4  ). 

As  a  consequence  of  equations  ((>),  the  determinant  <tik  ==-£•!,* 
and  \\c  ma\"  divide  the  S\  into  t\\'o  lainilies,  according  to  the  value 
of  this  determinant.  Hence  \ve  have  the  theorem: 

III.    <>lt    (I    nullfjui-iitll'ii'il    ijlltlil  /'(/•    <>t     tliliU'HKlUHX     -  />    ill     t(    xjittif    "/' 

mill    <U//H'/t#i<i>ix    '2  j>  +  \     thfi'c    iti'f    {«'<>  Ju/iit/tt'X    i  if  liiiiiir    njnti-fti    <>f 
</t//ti  tixtoiis  i>. 

No\\-  the  ('((nations  ol  anv  one  ,s'r  mi  (  I  )  mav  he  written  Itv  a 
proper  choice  of  coordinates  without  chan^in^  the  form  of  (4),  as 

»/,=  .'>  (/  =  !,  2,  «...  //-(-I)  (  7  ) 

In  fact.  \ve  have  simpK"  to  make  a  change  of  coi'u  dinates  h\ 
\\hidi  the  ri^ht-hand  nienibers  of  equations  (  .">  )  are  taken  etpial  to 
j~'(  and  then  t(>  drop  the  primes. 

(  'misider.  t  hen.  t  he  intersect  ion  of  (  7  )  with  anv  N'  wh<  >se  equations 
are  in  the  form  (.'))  wit  h  <itt  --  <•,  \\  here  >•  -  j  1  .  Then  (  ;>  )  is  ot  the 

*  Sri.ii'>  "Tlirury  i  if  1  )f[c  mi  in;i  ni-."  p.  l.'.T. 


404 


.V   DIM  !•:  X  SION  A  I,  ( i  K( )  M  KT 1  { V 


same  family  as  (7)  when  «J  =  1,  and  is  of  the  opposite  family  when 
(•-:—!.     The  coinlitioii  for  the  intersection  of  the  two  S'  is 


(8) 


If  [>  is  odd,  e(|uatioii  (  <s  )  is  satislied*  alwavs  when  *•=  —  1,  but 
is  not  satisfied  when  c  =  1  unless  other  relations  than  (  ti  )  exist 
between  the  coellicieiits.  If  p  is  even,  equation  (  <s  )  is  ulwavs  satis- 
lied  when  •  =  1,  hut  is  not  satislied  in  general  when  >•  —  —  1.  Hence 
\\'e  have  the  theorem  : 


IV.  If  i1  /*  ''/'  '"/'/  nn)iJnjr,  tiro  liitt'iir  x[i«<-<'x  ,V'  of  t>j>]>oxi'te  fawHics 
nn  it  tjHtitli'it'  tn  <i  xjHii'i'  of'  '2  j>  +1  diinenxioHx  iiht'itifx  inti'rui'ct^  <i/nt 
tifn  ,S''  "f  tin-  Kit/ni'  J  it  mil  i/  </"  li"t  //i  </i'/tt'/'<il  intc/'xi'ct,  if  I1  '•"<  <"'  <'''»'" 
ini/iJ'i  i\  t  n'n  ,S'  ,,('  tin1  xii/iti1  fitinilit  (tlu'difx  uiti'/'xi'i'f,  «/nl  tu'u  .S''  af 
iijijinxt/i'  Jitiniln'8  il"  ii"t  in  i/i  HI  fid  tnti'rxi'i-t. 

It  is  easily  shown  that  anv  point  /,'  on  (/)  mav  lie  ^i\'en  the  coor- 
dinates ^-=  0,  r-=  0,  (i-  \,  '2,  ••-,/>),  ti  .  j  :  ./•  _  ,  —  1  :  1  without 
chan^in^  the  form  of  the  equation  (4).  The  tangent  hypeiplane  7^ 
at  I[  is  then  ti  fi  M—  ./-y,  ,,  —  0,  and  its  intersection  with  $  is  the  >S'^'_j 


Anv  point  /!  on  this  locus  mav  he  '^iveii  the  coiirdinates  ut  =  '), 
./•  -  0,  (  /==!,  J,  •...//-  1  ),  »//(:  i//i  +  1:.r/(:.rI)  +  1  =  l:l:l:l.  The  line 
/;/.'  is  then  on  $.  The  tangent  hyperplane  to  (/)  at  /._'  is  then 
,.  i  :  "  iind  intersects  's>',",l_1  in  the  •vv~",1_., 


Anv  jiomt  /'  on  this  locus  can  now  lie  n'lven  the  coi'irdinates  ?/,=  ", 
./•  i).  (/-  1.  -2,  •  •  -./'-^),  w,.  .,:  »/J(:  //;,  .,:.r,  ,:./•;./;.  ,^  1  :  1  :  1  :  1  :  1  :  1  , 
and  the  N'  determined  h\-  the  three  points  /'.  /.!.  /.'  lies  on  (/>  and 
has  the  equations  //.=  •••  —  //  :  ./•  :...._;./•  :  I),  //  ,  —  j~  ,, 

•  '  -  '  -  ' 


POINT  COORDINATES 


40-J 


Proceeding   in   this  \v;iy  we   may  sho\v  that   unv  ,\'  (  /,'  •    y)   1\ 
on  $  c;iu  IK-  n'ivri!  the  equations 


without  cliaii^in^  tin-  I'nnn  of  equation  (  1  ). 

Anv  .V'  on  <p  has,   as  we  have  sri'ii,   the   equations  (•>),  and  if  it 
also  contains  all  points  of  ('•')<  its  equations  reduce1  to  the  form 


(10) 


where  the  coet'iicieiits  satisfy  conditions  similar  to  (  ii  )  and 

",.         '  '  '  ",.,.  t 

W=S|«     </."'//      ',=l'' 

\Vithout  change  of  the  form  of  equation  (  I  )  or  (  l>  )  anv  one  of 
these  ,S''  can  be  ^'iveii  the  etjinttions 

",    :  -'•,-  ( 1  -  ) 

In  fact,  we  have  simplv  to  make  a  change  of  variables  bv  which 
the  riidit-hand  members  of  equations  (  1<)  )  become  ./  '  and  then  to 
drop  the  primes. 

The  X'  !_M\en  by  (  1 '_' )  will  intersect  any  .s''  '4'i\cn  b\'  (  1  U  )  al\\  a\  s 
in  the  points  of  ,s';'  ^i\cn  by  ((.l).  In  order  that  (  1  '2 )  and  (  I<h 
should  intersect  in  some  other  point  not  in  N;'.,  it  is  m-eessai'v  and 
sut'ticieiit  that 


406  .V-DIMENSIOXAL  UKOMETRY 

Now  if  i'  —  k  is  an  odd  number,  equation  (1-V)  is  always  satis- 
tied  when  .  —  1 :  aiul  it'  p  —  1:  is  an  even  number,  it  is  always  satistied 
when  c  -  1.  Further,  we  notice  that  if  (\'l)  and  ^lU)  have  in 
common  a  point  /'  wliich  is  outside  of  N[,  thev  have  in  common 
the  N^. i  determined  by  >'{.  and  /':  and  since  (1-)  and  (10)  are  on  (/>, 
this  N^j  is  (jn  (/>.  .Moreover.  [>  —  k  is  odd  if  p  is  odd  and  k  even 
or  if  //  is  even  and  k  odd,  and  //  —  k  is  even  if  both  p  and  k  are 
udd  or  if  both  j>  and  k  are  even. 

From  this  we  have  the  following  results: 

1.  If  [i  is  odd  and  two  .s''  of  the  same  familv  intersect  in  an  S'k 
where  k  is  even,  they  intersect  in  at  least  an  >',[..,. 

±  If  //  is  odd  and  two  N'  of  opposite  families  intersect  in  an  S'k 
where  /•  is  odd,  they  intersect  in  at  least  an  X,'  ,. 

3.  It  /'  is  even  and  two  N'  of  the  same  familv  intersect  in  an  .s1,' 
where  k  is  odd.  thev  intersect  in  at   least   an  -^[.,r 

4.  \i  ji  is  even  and  two  .s1'  of  opposite  families  intersect  in  an  .s'x' 
where  /.'  is  even,  they  intersect  in  at   least  an   N,' , ,. 

This  mav  be  put  into  the  following  theorem,  with  reference  also 
to  theorem  IV: 

V.  If  l'  /•»'  '"A/,  t"'"  N';  "f  tin'  mi  in  I'fiiinHii  if"  ii"f  i/l  i/i-/ti'/;il  t'/tti't'- 
ni'i-f,  lut  imii/  iiit,'i-Ki-i-t  in  <in  X[.  irlt,  /-I-  /,'  /N  "</,/;  and  (/''»  ,S'^  nf  f>j>ji»x/ti' 
filniilifS  t/tt'-rvfi-t  in  i/'-ih  fil  in  a  fi'nnt.  fmt  ///<///  infet'St'ct  In  an  S'k  trh,  /•>• 
/.-  /.•>•  I'i'i-it.  If  I'  '*  <'''>'".  fir,,  S[  i ,f  tin'  X'tuii' f<i mi! ij  inft'/'tn'<-t  in  iji'in  i'<i! 
in  a  x!/t;/?i'  ji'i'mt.  Init  nl'i/l  tnt'  /'Xi'i't  tn  <in  .^'  whrfi'  k  i*  t'l'i'/l  ;  anil  tim 
S'  '//  <.>[i]nii<ltt'  tmnilti'X  d'.i  ii"t  in  i/>n> /'<(/  tntiTxi'i-t,  but  until  tnf>  I'^f't  in 
an  S't.  n'hi-rc  k  i*  <»ld. 

If  in  eijuatit'iis  (l(i)  \\ f  take  k  —  p —  \,  thev  reduce,  to 


with   '',,  =  »•  ~  _L  1 .     Hence  we  have   the  theorem: 

VI.  7'// /•«>/;///  mi  i/  N'    ,  on  (f)  //»  t//'"  S',  "/I,-  "f  t'lt/'li  famUif. 

Mme  geiierallv  the  number  nf  independent  cdetlicicitts  in  (  1"  )  is 
known   tVniii  the  thenrv  nf  determinants  tn  be  -  '  — 

I  lellre     \\'r     ha\'e     the    t  liei  d'elll  : 

VII.  T///'"/i>/lt  -/////  .s','  -//(  c/)//"  /: '  ' 


POINT  COORDINATES  407 

EXERCISES 

1.  Show  that  if  >','.  lies  on  </>  it   must   lit-  in  its  reciprocal  polar  space. 
From  that  deduce  the  condition  /•  —       f> 

r  •  '   , 

2.  Prove   that   there   arc  X    -     '  "'>','•  <iii   <£    by    determining   the 

number  of  solutions  of  equations  (  L* )  and  (M),  remembering  that  each  oi 
the  /•  -+-  1   points  may  be  taken  arbit  rarily  on  *'r. 

3.  Show  t  hat  through  every  >','.  Iv  ing  on  c^>  t  here  pass  -s.    -  S'r 
which  lie  on  the  qiiadric  ( /,-  <  /•  =      4>      j- 

168.  Stereographic  projection  of  a  quadric  in  Sn  upon  S,',  t.  Let  c/> 
be  a  quadric  hvpersnrlace  of  dimensions  n  —  1  in  N  ,  X  anv  hvper- 
plane  in  ,s'(,  so  that  X  is  an  N'  ,.  and  O  anv  point  on  (/).  Strain-lit 
lines  tlirou'j,'li  <>  intersect  <£  and  !i  in  general  in  one  point  each, 
and  set  up,  therefore,  a  point  correspondence  ot  (/>  and  ^.  winch 
in  general  is  one-to-one.  There  are.  however,  on  both  <£  and  1 
exceptional  points.  On  c£  the  point  <>  is  exceptional,  since  lines 
through  i)  and  no  other  point  of  c£>  lie  in  the  tangent  Ii\perj)lane 
at  < >,  the  intersection  of  which,  with  !£,  is  an  N'  .,  \\hich  we  shall 
call  TT.  lieiice  (>  corresponds  to  anv  ]>oint  of  rr.  ( )n  i  the  points 
in  which  the  straight  lines  on  (/>  tln'oii^'h  O  intersect  ^1  are  excep- 
tional, since  each  of  these  points  corresponds  to  an  entire  straight 
line  on  c/x  These  straight  lines  are  the  intersections  of  <£>  (>'',"',') 
and  the  tangent  livperplane  ( .V' ._,)  at  <>.  and  therefore  intersect 
^  (•*>','_,)  in  an  A'.';\,  which  we  shall  call  LI.  Kvidently  11  lies  in  TT. 

These  statements,  which  are  geometrically  e\  ident,  ma\  be  \  rrilied 
bv  the  use  ol  coordinates.  Let  ./•  :./',:•••:./',,.,  he  coordinates  ot  a 
point  in  \.  and  let  ,?+.,*+  .  .  .  4.^=0  (  1  ) 

be  the  c(|iiation  of  0.     \\'ithont   loss  of  o-eneralitv  we   mav   take  '' 


p.\\  ii  +  X.rr 

p.\  ,      0+X.' 

p  \  I   -f  \.l'  , 

fj.\  }-\  f  \./  . 


4US  .V   DIMENSIONAL   GEOMETKY 

and  (>/'  meets  i  in  the  point  (t>,  obtained  l>v  placing  A")(=0  in  ('2). 
This  determines  \,  and   the  coordinates  of  (t>  are  found  to  he 


where  £  arc  coordinates  of  points  in  !£,  and  ./•.  are  coordinates  of 
points  on  </>.  Since  ./•_  satisfy  equation  (1  )  \ve  may  write  the 
relation  between  /'  and  its  projection  <J  in  the  form 


pc>n  -•'•„  .,>=£,-+  £;  +  •  ••  +  f;  ,- 

Kijiiiitions   (  •>  )   sliow   that    to  a   definite   point    /'  corresponds  a 

definite  point  <t>.  except  that  the  point  <>  ^ives  an  indeterminate  (J 
on  the  locus  ^  —  0.  which  is,  therefore,  the  equation  of  TT  in  —  . 
Also  any  point  (t>  corresponds  to  a  definite  point  /',  except  that 
any  point  in  the  locus  £„  —  0,  £~  +  £.;  4-  •  •  •  -f  f,';'.  ,  =  (l  i;'i\'es  an  inde- 
tenninate  point  /'.  but  such  that  /'  and  n  lie  on  a  straight  line 
tliroii^'h  <  >.  Therefore 

L  =",    ff+IJ+---+f;.,=  o  (4) 

arc  the  equations  which  define  the  qiiadric  £1.     \Ve  may  note  that 

any  point    ','  which  is  on  TT  but  not  on  LI  LM\CS  the  definite  point  <  >. 

Any    >',.'    \vliich    lies    on    (f>    pi-ojeets    into    an    >','   on    X.     For    the 


An  >';'  on  (f>  intersects  t  he  tangent  hyperphun1  at  o  in  an  .\'  ,  whieli 
pi-ojects  into  an  S'  ]  in  i.  I>ut  all  points  of  the  tangent  hvper- 
plaiie  project  into  points  on  11.  and  therefore  this  N;'  ,  lies  entirely 
i  in  £2.  Therefore  we  say  : 

7.     /!>/   xfi-i'f";li'itpJiif  /'/•';/'•'•//'-//   tiiuj  1  nii'it  r  >y/'V  ,S';!   //////'/  "//   ii   ift/'iJ- 

/•/',•     Jll/n>    ,•*!'  /'t  </<•!       (f)      III      'I      KI'I!''!'      "I       II      lllllli   IIX/"/IX      I  ft      f>r<>U'//lt       Illt'i      i':>/'/','- 

viiuiiili'ifi'   n'/tli   <t   hiii'ir  ftinii-i-    S't     ]    li/m<i  "it   it   utitti/t'ti'   anrtiti-i'   LI   in 

'/     >•//./<•/       til      il  '_'    ll  i  illi'llfiollK. 

'\'\\\^  beiii'^  jii'oyeil.  let  us  coilsidel1  the  case  ill  which  //  is  an  odd 
n  umber  '2  /'  -*-  1  .  Then  $  j>  of  dimensions  '2  />.  and  LI  is  of  rlimensions 


POINT  COORDINATES  400 

2  j>  —  '2.  ( )n  0  there  exist  linear  spaces  A';'  which  project  into  linear 
spaces  ot  the  same  niiinlier  of  dimensions,  which  we  call  i.'(  since 
thev  are  in  ^i.  An\'  two  ^L'  intersect  in  at  least  a  point,  since  thev 
lie  in  a  space  of'l!/>  dimensions  (^1o'4).  If  that  point  of  inter- 
section is  not  on  12,  it  corresponds  to  an  intersection  of  the  two  A';'  on 
0,  since  outside  of  12  anv  point  of  X  corresponds  to  a  definite  point 
of  0.  If,  however,  the  intersection  of  two  X'  lies  on  12,  the  two 
corresponding  A'  on  0  do  not  in  general  intersect.  In  fact,  the  in- 
tersection of  two  ^ '  on  12  simplv  means  that  a  straight  line  from 
<>  in  the  tangent  hvperplane  at  <>  meets  each  of  the  two  correspond- 
ing A''.  Since  we  are  talking  of  two  A'  in  general,  their  intersection 
in  the  tangent  hvperplane  at  <>  mav  he  considered  as  exceptional, 
si  i  that  we  have  the  the<  >ivm  : 

II.  />  1  H'o  A''  nil  tin'  tjU<lil/'H'  0  (ntt>.rni'i-t.  (In'  rnf/'fHjiHHiftitt/  A'  }  on 
tin  ijiiitilrii1  12  </o  not  in  i/f/n-ntl  intf/wct  ;  and  //'  tiro  A'  on  0  do  not 
tiitt'i'Xfi't,  tint/'  f(i/'/'rnj>t>ndiHi/  A''_1  on  12  in  iji'iit'i'itl  tntf/'Xfft  in  ti  j/ot/tt. 

In  a  similar  manner  the  question  of  the  intersections  of  linear 
spaces  A'  ,  on  an  A'.,'1].1  .,  may  he  reduced  to  the  question  of  the  inter- 
section of  t  \\  o  A'  .,  on  an  A,-1  and  eventually  to  the  intersection 

/        -  -  /•  - 4' 

of  two  Aj  on  an  A.,-1:  that  is,  of  two  straight  lines  on  a  quadric  sur- 
face in  ordinarv  three  space. 

We  mav,  accordingly,  divide  the  A'  on  0  into  two  families,  accord- 
ing as  thev  correspond  l>v  this  successive  projection  to  the  two 
families  of  generators  on  an  ordinary  quadric  surface.  From 
theorem  II.  however,  it  is  evident  that  we  have  the  same  classi- 
fication as  that  made  algehraically  in  vj  lt>7:  for  it  follows  that 
two  A'  of  the  same  familv  do  or  do  not  intersect  according  as  /<  is 

even   or  odd,  and  two  A'  of  opposite  families  do  or  do  not   intersect 
/•          i  i 

according  as  /<  is  odd  or  even.  Exceptions  mav,  of  course,  occur, 
as  has  1  iceli  shown  in  vj  1  'i  i  . 

Let    us  consider  now  the  intersection   of  0  l>\    anv    hvperphuie 

uhidi  passes  or  docs  not  pass  through  the  center  of  projection  i >. 
according  as  A/  4  ,/  ,  is  or  is  m>t  o.  The  intersection  \\ith  (/>  is 
an  A'-  \\liich  projects  upon  ^  into  ;i  i. ; '  ,,  \\ith  the  eijiiation 

(   "'      l     '',•:'  (  S  ,"  ~H    '   '   '      ^   S  ,      ;   *          '-  ''  \%\<*n      -    •    •   •  -  '',.      iS         i?.. 

(    ''-/  '  I        .  )£  (l. 


410  A'-DIMEXSIONAL  GEOMETRY 

This  is  in  general  ii—,,"!.,  \\hich  contains  12,  but  if  /</;i  +  <fn  ^  =  0, 
it  splits  up  into  tlir  hyperplane  TT  and  a  general  hyperplaiie  7. 
1  leiice  tlit1  theorem  : 

If  nn  N1,"1*  "/'""  $  '^"'s'  »"f  /"'-s'-s'  thrntit/h  <  >,  it  /</•*;/»•< -tx  info  a  <p«nlri<' 
in  ^i  H'/tii-h  ••"iititin*  11 :  //'  itn  N1^.,  ""  <£  <l<x's  pass  thruiinh  < >,  it  projects 
info  ,(  hyper  plane  in  ~  together  icit/t  tin1  hj/jK'rjila/tt'  TT. 

EXERCISES 

1.  Slio\v  that  any  >';',  on  <$>  not   passing  through  <>  projects  into  a  2^ 
in  2  which  intersects  TT  in  a  2;;,.  ,  contained  in  12. 

2.  Show  that  anv  >',',    ,  not  passing  through  <>  intersci-ts  </>  in  an  -s',;-° 
\vhich  in'oiects  into  a  2;'  ..  which  passt-s  /  times  throiiu'li  12. 

i          j  7i  —  _  i  ri 

169.  Application  to  line  geometry.  Sim-o  lint'  c-oiirdinatt'S  con- 
sist of  six  homogeneous  variables  connected  bv  a  (juadratic  rela- 
tion, a  straight  line  in  ordinary  space  niav  be  considered  as  a  point 
on  a  qnadrie  surface  in  an  .V..  \\'t-  shall  procrcil  to  interpret  in  line 
'^eonietrv  some  of  the  general  results  \ve  have  obtained.  In  so 
ddiii'j,'  \ve  shall,  to  avoid  confusion,  designate  a  point,  line,  and  plane 
in  N-  by  the  symbols  \',  .s'[.  .s'.',  respectively,  reserving  the  words 
"point,"  '  line,"  and  plane"  for  the  proper  configurations  in  -V,. 
Let  $>  be  the  (jiiadric  whose  equation  is  the  fundamental  relation 
connecting  the  coordinates  of  a  straight  line.  Then  an  N'  on  0  is  a 
straight  line,  an  S'  on  0  is  a  pencil  of  straight  lines,  and  an  A''  on  <£ 
is  either  a  bundle  of  lines  or  a  plane  of  lines.  These  statements  art- 
established  bv  comparing  the  analytical  conditions  for  pencils  and 
bundles  of  lines  given  in  ij  1ol  with  those  for  S'}  and  N.'.  on  (f). 

The  two  families  ot  .s'.!  on  0  are  easily  distinguished,  the  one 
consisting  of  lines  through  a  point,  the  other  of  lines  in  a  plane. 
It  is  evident  that  t\\o  ,s'.'  of  the  same  family  intersect  in  an  >>'.',  for 
two  bundles  of  lines  or  two  planes  of  lines  have  always  one  line  in 
common.  ( )n  the  other  hand,  a  bundle  of  lines  and  a  plane  of 
lines  do  not  in  general  have  a  line  in  common:  that  is.  two  ,s''  ot 
different  families  do  not  in  general  intersect.  If.  however,  a  point 
ot  lines  and  a  plane  of  line-;  have  one  line  in  common,  they  \\ill 
have  a  pencil  in  common  :  that  is.  it'  fir,,  .s'.'  nf  ://t}\  rr/if  fiimilit'S 
"/i  (£>  !nt>  i\t>  <-t  in  iin  N'.  tin  >i  i///'  /•>••  •<•/  in  ii/i  N,'.  'I  his  is  in  accord 

\\  ith    theorem    \'.    ^   1  1 17. 


POINT  COORDINATES  411 

A  linear  line  complex  is  an  >S'a-'  formed  by  the  intersection  of  c/> 
and  an  S'4.  If  the  Nj  is  tangent  to  $,  the  complex  is  special  and  con- 
sists of  x'.s'l  joining  the  points  of  the  complex  to  a  iixed  A','.  The 
special  linear  complex  in  line  geometry  consists,  therefore,  of  f.~ 
pencils  of  lines  containing  a  fixed  line. 

A  linear  line  congruence  consists  of  an  .S'.V"'  formed  by  the  inter- 
section of  (/>  and  two  N|.  Therefore  it  consists  in  general  of  lines 
each  of  which  belongs  to  two  pencils  containing,  respectively,  one  of 
two  fixed  lines.  When  the  two  Iixed  lines  intersect,  the  congru- 
ence splits  up  into  a  bundle  of  lines  and  a  plane  of  lines,  with  a 
pencil  in  common.  That  suggests  the  theorem  that  on  </>,  //'  the 
tii'u  fixed  Ay  connected  icith  a  cont/ruence  >S?rJ  lie  on  an  *S'[  of  0,  the 
A*'.'"'  sjilitti  itj>  into  ttco  .s'o  <>f  different  families  intensectin;/  in  (hi*  N,'. 

A  linear  series  is  an  ,s'i'J)  determined  by  the  intersection  of  (f>  and 
three  >S'^.  From  the  general  theory  we  see  that  the  series  consists 
of  jil  lines,  each  of  which  lies  in  a  pencil  containing  each  of  x1 
fixed  lines.  It  therefore  consists  in  general  of  x1  lines  intersecting 
another  cc1  lines.  \Ve  leave  to  the  student  the  task  of  considering 
the  special  cases  of  a  line  series. 

A  linear  complex 

ar>\  +  a.,j\,  +  •  •  •  +  <tn  + !  .r ,  + !  =  0  (1 ) 

is  fully  determined  by  the  ratios  al  :  a.2:  •  •  •  :  alli.l,  which  may  be 
taken  as  the  coordinates  of  the  complex,  and  we  may  have  a 
geometry  in  which  the  line  complex  is  the  element. 

The  quantities  a^  :  a., :  •  •  •  :  an  M  are  also  the  coordinates  of  a  point 
in  N-,  which  is  the  pole  of  the  hyperplane  (1).  Therefore  the 
point  <if  is  not  on  the  quadric  (/)  unless  the  complex  is  special.  An 
,S''  in  ,S'.  is  therefore  a  line  complex.  The  lines  of  the  complex  <i( 
correspond  to  the  points  in  which  the  polar  (1)  of  the  point  a{ 
intersects  (f).  If  ,S^  is  on  (/>,  the  complex  is  special  and  mav  be 
replaced  by  its  axis  so  as  not  to  contradict  the  previous  statement 
that  an  ,S''  on  0  is  a  straight  line.  In  fact,  if  the  equation  of  (/>  is 
taken  as  ^.'7  =  0,  the  coordinates  of  a  special  complex  and  oi  its 
axis  are  the  same. 

( 'onsider  now  two  complexes  tt:  and  !>t  as  two  points  N'  in  Np. 
They  are  said  to  be  in  inr<i]uti<>n  if  each  X'  lies  on  tin1  polar  plane  ot 
the  other.  From  this  it  follows  at  once  that  if  one  of  the  complexes 


412  .Y-DIMEXSIONAL  (JEOMETRY 

is  special,  its  axis  is  a  line  of  the  other;  so  that  if  both  are  special, 
their  axes  intersect,  and  conversely.  In  case  neither  complex  is 
special,  the  >','  defined  by  a,  and  l\  are  not  lines  in  .Vt,  and  we  must 
look  for  other  geometric  properties  of  complexes  in  involution. 

In  N,  the  coordinates  </,  and  A.  have  a  dnalistic  significance.  On 
the  one  hand,  they  are  coordinates  of  two  ,s',' ;  on  the  other  hand, 
thev  are  coordinates  of  two  hyperplanes,  the  polars  of  these  points. 
The  two  N'  determine  a  pencil  of  .S'^  which  lie  in  an  Np  and  the  two 
hyperplanes  a  pencil  of  hyperplanes  which  have  an  /\  in  common. 
The  pencil  of  N'  contains  two  .s'^  on  $,  and  the  pencil  of  hyperplanes 
contains  two  hyperplanes  tangent  to  $.  It  is  then  evident  that 
ftt'"  cowph'ft'S  (ire  in  involution  tchcn  t/ie  ttco  S't  in  <S'6  which  /vy//v.s>-/^ 
than  arc  harmonic  conjugates  with  respect  to  the  qnmlrlc  <£,  or,  what 
is  the  same  thing,  when  the  two  hyperplanes  defining  the  com- 
plexes are  harmonic  conjugates  to  the  two  tangent  hyperplanes  to 
0  which  arc  contained  in  the  pencil  defined  bv  the  two  complexes. 

It  is  clear  that  in  any  pencil  of  complexes  the  relation  between 
a  complex  and  its  involutory  complex  is  one-to-one. 

If  we  consider  a  fixed  complex  </_,  all  complexes  in  involution  to 
it  are  represented  by  points  in  an  N4,  which  is  the  polar  hyperplane 
of  <r  with  respect  to  (f). 

This  relation  can  be  generalized.  Let  N/.  be  a  linear  space  of 
points  in  .s'.,  and  let  >^_t.  be  the  conjugate  polar  space  with  respect 
to  $,  so  that  any  point  in  .s1^.  is  the  harmonic  conjugate  with  respect 
to  0  of  any  point  in  ,S4 _t.  We  have,  then,  two  scries  of  complexes, 
each  of  which  is  in  involution  with  each  one  of  the  other  scries. 
The  points  in  which  N[.  intersect  (f>  are  special  complexes.  Their 
axes,  therefore,  must  lie  in  each  of  the  complexes  in  X4'_|.,  as  has 
been  shown  above.  In  other  words,  the  </.rr.v  <>f  th>-  xjH'i-iil  t'umjtlejri'ts 
nt  ant-  xt'/'ifx  ii/'i'  t/n'  xtmii/ht  liiii-x  fiini ui'in  ><>  the  I'oiiipli'Xt'H  of  ttie 
int'vlut»rij  .sc/'/cx,  (i/dl  i-ij/n'i'fyt'li/.  I  he  prool  ot  the  converse  is  left 

to   the    Student. 

For  example,  consider  the  pencil  of  complexes  <it  +  \?>t  in  invo- 
lution with  the  series  of  complexes  ^  -f-  \'</i  -\-  /zV(.+  v  1\.  The  pencil 
of  complexes  have  in  general  a  congruence  of  straight  lines  in 
common,  and  these  tire  the  axes  of  the  special  complexes  of  the 
series.  On  the  other  hand,  the  series  of  complexes  have  in  general 
two  lines  in  common  which  are  the  a\cs  of  the  special  complexes 


POINT  COORDINATES  413 

of  the  pencil.  Again,  consider  the  bundles  of  complexes  <it  +  \f>:  -f  ^ 
and  f, -t-XY,-)-  ////,  in  involution.  The  complexes  of  cither  bundle 
have  in  common  the  x1  straight  lines  of  a  regains  which  arc  the 
axes  of  the  special  complexes  of  the  other  bundle. 

Any  eollineation  of  .V  is  a  transformation  of  X,  by  which  a 
linear  line  complex  goes  into  a  linear  line  complex,  and  any  linear 
series  of  complexes  goes  into  another  such  scries.  If,  in  addition, 
the  quadric  $  is  transformed  into  itself,  straight  lines  in  X,  are 
transformed  into  straight  lines,  and  any  X!  on  $  is  transformed 
into  another  X.'  on  (f>.  But  as  there  are  two  systems  of  X'  on  $, 
the  transformation  may  transform  an  X.[  either  into  one  of  the  same 
system  or  into  one  of  the  other  system.  In  the  lirst  case,  points 
in  Xj  are  transformed  into  points;  in  the  second  case,  points  in  Xs 
are  transformed  into  planes.  We  have,  accordingly,  the  theorem  : 

A  '•ollini'ittion  in  X  irJii'-h  !<><im<  tJn'  ^miilri'  c/>  unxlfcrfif  /x  rit/nr 
a  cvllineation  <.>r  n  correlation  in  X',. 

EXERCISES 

1.  Discuss  oriented  circles  in  a  plane  as  points  on  a  quadric  in  >'4. 

2.  Discuss  oriented  spheres  in  ordinary  space  as  points  on  a  ijuadrie 
in  X.,. 

170.  Metrical  space  of  n  dimensions.  We  have  been  considering 
spaces  in  which  a  point  is  defined  by  the  ratios  of  homogeneous 
variables.  We  mav.  however,  consider  equally  well  a  space  in 
which  the  point  is  defined  directlv  bv  n  coordinates  »  ,  ?/,•••,  ?/  , 

I  .  I          2  « 

and  where  the  equations  are  not  homogeneous.  All  equations  mav 
be  made  homogeneous,  however,  bv  placing 


The  discussion  is  then  reduced  to  the  homogeneous  ease,  but 
the  use  of  t  as  the  n  +  1  *("  coordinate  emphasizes  the  unique  char- 
acter of  that  coordinate.  In  fact,  when  /--<),  sonic  or  all  of  the 
original  coordinates  become  inlinite.  This  enables  us  to  handle 
infinite  values  of  the  original  coordinates.  Such  sets  of  \alues 
mav  be  distinguished  from  each  other  bv  the  ratios  of  ./• .  so  that 

</:</:•••:</    :  0 


414  A"  DIMENSIONAL  (JKOMKTHY 

is  said   to  define  a  definite  point   at   infinity.     \Vc  liave,  therefore, 
a  speeial  ease  of  protective  space  with  a  unique  hyperplane  f  =  ". 
\Ve  mav  define  a  distance  in   a  manner  analogous  to  that  used 
in  three  dimensions,  by  the  equation 

'/"'  =  (  "i  -  ",  )•  4-  (  ».!  -  a.,  )  '  4-  •  •  •  4-  (  u'n  -  "„  )% 
or,  in  homoeneous  form, 


From  this  it  appears  that  the  distance  between  two  points  can  be 
infinite  only  if  f  or  ('  is  zero.  Conversely,  with  the  exception  noted 
below,  a  point  for  which  f  —  0  is  at  an  infinite  distance  from  any 
point  for  which  t'  —  0.  Therefore  t  =  ()  is  called  tJn-  hyperplane  <if 
hi  ti  nit  if. 

On   the    hyperplane    at    infinity    the    coordinates    are    projective 

coordinates   in    N         defined   by   the    ratios   .r  :./•:•••:  ./•  . 

i      -j  11 

An  exception  to  the  statement  that  points  mi  the  hyperplanc  at 
infinity  are  at  an  infinite  distance  from  points  not  on  that  hyper- 
plane occurs  for  points  on  the  locus 

t  =  0,      j-f  +  •'•;  +  •  •  •  4-  f-  =  0,  (  4  ) 

since  the  distance  of  any  point  on  this  locus  from  any  other  point 
is  indeterminate.  This  locus,  which  is  an  N"  .,,  or  a  quadric  hyper- 
surface  in  the  hyperplane  at  infinity,  is  called  t/f  ///•>•'</>//.•. 

The  following  properties  of  metrical  space  are  such  obvious 
generalizations  ol  those  of  three-dimensional  space  that  a  mere 
statement  of  them  is  sufficient. 

A  ~hypiT*p1u're  is  the  locus  of  points  equidistant  from  a  fixed 
point.  Its  equation  is 

(  ./',  —  ",  )':4-  (  •'•„—  ".,  ):4-  •  •  •  4-  '.  ./•,'•    -  "„  f  =  r,  (  •">  ) 

and  it  is  obvious  that  all  liyperspheres  contain  the  absolute,  but 
no  other  point  at  infinity. 

A   straight   line  may  be  defined  hv  the  cijuatioiis 

./•     -  'i 

•••     '    "' 


POINT  CWUDIXATKS  41-") 

the  ratios  (if  tlu1  quantities  /.  We  say  that  these  <|iiantit  ics  deter- 
niiiit'  the  direction  df  the  line,  direction  bein^  that  property  which 
(list  in^uishes  between  straight  lines  through  the  same  [mint. 

Two  lines  with  the  same  direction  meet  the  hyperplane  at 
infinity  in  the  same  point  and  are  called  parallel.  Two  lines  with 
directions  /,  and  /'  meet  the  hvpcrplane  at  inlinitv  in  two  points 
with  coi'irdinates  li  and  /',  and  the  straight  line  connecting  these 
two  points  nit'ets  the  absolute  in  two  points  >uch  that  the  cross 
rat  io  of  the  four  points  is 

Vi  +  7/l+  •••  -H/' 

\  (  /f  +  /;+•••+  /,-;  )  \  l[-  +  I','  +  •  •  •  +  I': 

We  shall  detinc  this  as  the  cosine  of  the  an^le  between  the  two 
lines:  namel,  ' 


In  particular  two  lines  are  jterpcndicnlar  \\hen 

/,/i+/a/s'+  ...  +/,/,;-". 

A    line   meets  the  absolute  when,   and  only  when, 

/?+/.;+  ••  •  -f  /,;=<). 

In  that  case  the  distance  between  anv  two  points  on  the  line  is 
zero,  and  the  line  is  a  minimum  line.  Through  anv  point  of  space 
L,ro.  then.  -s_"  '  minimum  lines  forming  a  hypereone  of  ~s.n~l  points. 

A  tangent  hyperplane  to  a  hvperspliero  intersects  it  in  x"^!  lines. 
and  since  the  sphere  contains  the  absolute  these  are  minimum  lines. 

Anv  hyperplane 


h  •  •  •  4-  "„.'•„  =  <\ 

\\iiii-h  is  a  hvperplane  in  the  .s'  .  delineil  b\-  /  -  -  <>.  It  is  tangent 
to  t  lie  absolute  \\  hen  N  ,^-  =  n. 

I  I  \  pci'plancs  sat  isfviiiLT  t  his  condition  arc  minimum  hypcrplanes  ; 
all  othei's  arc  ordinary  hvperplanes. 

Tlie  intersection  of  an  ofdinarv  hyperplane  with  t  »  has  a  pole 
witli  rcsjiect  t(.)  the  al.isolute  whose  coi'irdinates  are  -',:-'  :•••:''.. 


410  .V  DIMENSIONAL  (JKOMKTKY 

and  any  straight  line  \villi  the  direction  d::  a.2:  •  •  •  :  <in  is  said  to  l>o 
perpendicular  to  tin-  hvperplane.  In  fact,  from  tlic  definition  of 
perpendicular  lines  already  given,  this  line  is  perpendicular  to  any 
line  in  the  hvperplane,  and  conversely. 

Two  hvperplanes  are  perpendicular  when  the  pole  ot  the  trace 
at  intinitv  of  either  contains  the  pole  ot  the  trace  ot  the  other. 
Therefore  the  condition  for  two  perpendicular  hvperplanes  is 

„/,-!_„/,  4-  ...  +  „  1'  —  0. 

1    1    '     •    8    3    '  "    " 

It  follows  that  the  n  hvperplanes 

./-,=  0,     ./-.,=  <>,...,     .,-„  =  0 

are  mutually  perpendicular  hvperplanes  intersecting  at  (>.  Through 
<>  or  anv  point  of  space  pass  an  intinite  number  of  such  mutually 
orthogonal  hvperplanes:  for.  as  seen  in  £  1<>~>,  we  niav  lind  in  t  =  <l 
an  intinite  number  of  coordinate  svstems  such  that  the  absolute 
retains  the  form  "N  j-f  ~  0,  and  the  lines  drawn  from  n  to  the  points 
./•  —  0,  j-k  —  0  (Jc  '+-  i)  determine  the  hvperplanes  required. 

In  this  way  anv  ordinary  hvperplane  niav  be  made  the  plane 
j-n=  0.  The  coordinates  in  this  hvperplane  are  ./-,  :  ./•.,  :  •  •  •  :  ./;.  ,  :  /, 
and  its  absolute  is  t  —  Q,  ./y  +  ./•.  :-(-•••  +./-;_1  =  0. 

Tlieret'ore  f/ir  <i<'<>nii't  r;i  in  ituij  <//•<////<//•//  liifpcrplitm1  <l!lY<'i'x  fi'mii 
tJiitt  in  f/n'  unijiind  xjnii-1'  <>nlij  In  tin'  niniJit  r  '_//'  tin'  (liiiit'nxinnx. 

Two  linear  spaces,  S'r  and  S'r  ,  are  said  to  be  completely  /»>r/tUt/ 
if  they  intei'sect  only  at  intinitv  and  if  the  section  of  N.'  at  intiniiy 
is  completely  contained  in  the  section  of  S'r  at  intinitv  ('*.  —>*.,}• 
Since  the  section  of  V  at  intinilv  is  an  ,s-'  _,.  it  is  necessarv  that 
N,!  and  .s','  should  lie  in  an  >>','  ,_,.  _,,._,,—  N,'.  ,,  (theorem  I\'.  ^j  1»>1). 
Morenver.  if  we  take  /^  points  in  the  ,S',!  _,  at  intinitv,  one  other 
point  not  at  intinitv  in  S'y,  and  /•,  —  /•  -4-1  points  not  at  inlinitv  in 
S'r  ,  we  have  r+  -  oints  to  iletermine  an  .s''.  .  Therefore. 


Consider  now  two  spaces.  .N''  and  S'r  (  /•  ":/•.).  \\hicli  do  not  in- 
tersect (/'  |  +/•.,••  a).  Tlicv  determine  in  the  hvperplane  at  intinitv 
two  noiunt  erscrt  in1.;'  spaces.  V  ]  and  .s'.'  ,.  If  we  take  /•  points  in 
•s','  _,.  and  /•.,  points  in  S'r  ,.  we  detennine,  bv  means  ot  these  points, 

an   N'         ,   in   the   hvperplane  at   inlinit\"  which  contains  both   S' 

.-•  .  i      i  i  '  ,  —  i 


POINT  rooKniNATEs  417 

and  .S''          I»v  means  of  this  ,S''         .  and  one  other  point  in  ,N'   not 

'  •„>     '  '  i       •;     '  i 

at    infinity,   we  detonnino  an   .V'     .    which   contains   ,s''    because   it 

IS  '  1 

contains  ^4-1  of  its  points,  and   is  parallel  to  ,s''    since  the  inter- 
section with  intinitv  of  N'    is  completely  contained  in  that  of  N'  . 

''•:  •  i        i 

Hence, 


i«'<'fin</  li/ifur  ximi-fx  >r/fli  r  ~  r  ,  it 

^ 

is  pt>ttftif>/f  t»  iniss  tt  litii'ti/'  ,<••/""'''  N'         thi'nniih   S'    iixrdllt'l  t<>  S'  . 

i  r  ':  i   J  i 

It  is  obviously  possible  to  define  as  partially  parallel  two  linear 
spaces  which  intersect  at  infinity  and  nowhere  else.  This  would 
lead  to  a  series  of  theorems  of  which  those  in  ^  1->X  are  examples, 
but  we  shall  not  pursue  this  line  of  investigation. 

Two  linear  spaces  will  be  defined  as  completely  perpendicular 
when  each  straight  line  in  one  is  perpendicular  to  each  straight 
line  of  the  other.  If  S'r  and  ,V'  are  two  linear  spaces  intersecting 
the  hyperplane  at  infinity  in  /\'._i  «uid  A','.  .r  respectively,  it  follows 

that  the  necessary  and  sufficient  condition  that  .V'.   should  be  com- 

i 

pletelv  perpendicular  to  X'  is  that  S'.  ,  should  lie  in  the  conjugate 
polar  space  of  .S'?'  _l  Avith  respect  to  the  absolute,  when,  of  course, 
.S''  _T  will  also  lie  in  the  conjugate  ]>olar  space  of  -s','.  „,  with  respect 
to  the  absolute. 

Now  the  conjugate  polar  space  of  S'r  in  .V'  ,  (the  hyperplane  at 
intinitv)  is,  by  Jj  1<>5,  »S^_r  _,.  If  ,S','  is  given,  its  intercejit  on  the 
plane  at  infinity  ,V'  ,  is  determined,  and  the  reciprocal  polar  space 
X'  r  _,  is  also  uniijuelv  determined.  (  )ne  other  point  in  finite 
space  then  determines  with  this  V  t  an  ,V'  ,.  which  is  completely 
[KM-pendicular  to  the  given  N  '  .  Hence  the  theorem. 

T/l/'<>>ff//t      III!!/     jiolllt      ill      KfHll'l'     lit/I'     (Onl     Ilillll     ntll'      ,S;'        r     I'lIU     fll'     jlllXXti) 

K'/iii'/i  /s  r/i)/ij>/t'tf/i/  ]if/'i>i'  n<l  it'iiliir  fa  it  i/ii'i'H  S'r.  Ani/  Inii'iti'  sjxii't' 
i-'ii/tii  i  ih'il.  iii  S'n  r  is  f  //>•//  I'niii  >>l>'til  i/  in  rin'tnl  I'-iil'tr  in  /in//  1  1  n>  't  r  sjiiiri' 
in  S'r. 

It  is  possible  to  define  as  partially  perpendicular,  spaces  each 
ot  which  contains  a  straight  line  perpendicular  to  the  other,  as  in 
:?  li)t!,  but  we  shall  not  do  this. 

Let  us  consider  the  stereograph  ic  projection  of  a  hvpersphere  upon 
a  hyperplane.  Mere  we  have  merely  to  use  the  results  of  £  lo'S, 
interpreting  the  ijuadric  </>  as  a  hypersphere,  and  the  plane  .rn  r  ^  —  () 


418  .V  -DIMENSIONAL  (IKOMKTHV 

as  tlic  liyperplane  at   intiiiity  in   N(1.     Then  TT  is  the  liyperplane  at 
infinity  in  N,    ,,  and  11  js  t)u»  absolute.    We  have  at  once  the  theorem  : 

/>'//  the  xtereni/fii  pJnc  projection  nf  it  nintersphere  in  Sn  upon  <i 
hilperpbnie  .\  ,,  hyperphtnnr  xf<'ti«nx  <>t  (£>  </»  into  hifperplnnen  <>r 
Jiiipi-rxpJti'rt'x  lit'  Sn  ,  iii'i'urtlhif/  iix  tin'  hifperplanar  xi'rfinHx  »f  (j)  >1<>  or 
«/"  nut  1'untiiin  tin'  fi'/ifi-r  nf  rnt't'twn. 


A  collineation  in  Sn  bv  which  (f>  is  invariant  skives  a  point  trans- 
formation on  (f)  bv  wliich  hyperplanar  sections  L;-O  into  liyperplanes. 
There  is  a  corresponding  transformation  in  tia_}  l>v  which  a  hvper- 
jilane  or  a  hypersphere  s^'oes  into  either  a  liyperplane  or  a  hvpersphere. 
It  the  collineation  in  Sn  leaves  (>  as  well  as  0  invariant,  hvper- 
planes  of  NM  ,  are  transformed  into  liyperplanes,  and  the  transfor- 
mation is  a  collineation.  lint  the  transformation  in  <S'(J  leaves  the 
tangent  liyperplane  at  o  unchanged,  and  therefore  the  correspond- 
ing transformation  in  \_,  leaves  the  absolute  unchanged.  Heiu-e, 

(_1<>?/hn'tit(ttnst  in  Sti  tr/i/'r/i  1,'iin-  (f)  it/I*/  ///I-  [H'int  O  <>n  (f)  unt'hanfied 
il>  ti-rnii  in:  rnlli  iii'iitmiix  in  ,V(  ]  H'JiicJi  It'iiri'  the  itlasolutc  unt'hfinffed  find 
u'hii'h  ni'i'  tliert'fnre  nit'triml  tnuixfurmntinnx. 

(  'nlli  ni'<itin))x  in  Sn  icJttfJi  /I'ld-i-  (f)  fiiif  not  i)  unchanged  <1*'t  ermine 
l«>iitt  trnnxfortnutii'tnn  in  S:i  ,  /<//  u'hic/t  lii/perxpherex  <i«  into  Junter- 
ftpJierex,  it  Jiifpei'plnne  ln'in</  ermxiilered  it  ftper'mt  i-iixf  <>f  n  InfperspJu're. 

\Ve  have  used  in  ^  1f'»S  one  set  of  coin-din  at  es  (.rt)  foi'  the  points 
of  </).  and  another  set  (  ^,  )  for  the  points  of  ,s';i  r  but  clearly  the 
coi'irdi  nates  ./•.  mav  also  be  used  to  determine  points  in  Sn  ,. 

\\"e  shall  have,  then,  for  the  points  of  ,S';  ,>/  +  1  lioinogenoous 
coi'n'dinates  connected  bv  a  quadratic  relation,  and  such  that  a 
linear  equation  between  them  represents  a  hvpersphere  with  the 
hvperplanc  as  a  special  case.  Kach  of  the  coordinates  .r(  equated 
to  /ero  I'cpresents  a  hvpersphere.  \Ve  mav.  accordingly,  call  them 
(//-}-  1  )-polvspherical  coordinates  of  the  points  of  N  r  Thev  are  a 
generalization  of  the  pentaspherical  coi'u'dinates  of  .s'..  We  say: 

I  '  r«jii'ti  *'>'   cniii-iHiKitett   of  points  <>n   n   liifperspliere   in    .S>'.(    <ir>'  polij- 

UnJll'/'ifill    fiil'il'lli  ll/tfi'X     nt'  jinitlfs    nil     illl     X        ,     ilttn    ll'llirll     fill      Jl  l/j>  iT*/  >//!'/''' 
/.v      nff/'i  -in/Til  lihi'-illl  If      llfuft'fff'll,       (  'nlli  lli'iltiniix       nf     Sf       ll'hli'h       liilt'i'      (In' 

I"//"  i'x/  '//••/•'•    i  ii  I'll  riii  nt   iit'i1   /if/i'iir  ft'<tHKt''if'inttti"H*  ';'    tlo'  p'ili/apjn'ri''ill 


POINT  rooKDINATKS  .j  ]'.i 

171.  Minimum  projection  of  Sn  upon  Sn    ^     (  'on.-ider  in  ,s  .  \viih 
nonhomogeneous  metrical  coordinates,  the  minimum  hvpereone 


The  section  of  this  l>v  the  hvperplane  ./•„  -    n  i 

•  •  •  +  (.'•   .—  " 


which  is  a  hvpersphere  in  the  N(  ,  defined  by  ./•='!.  \\V  sav  that 
the  vertex  '',  of  the  minimum  hvpereone  (  1  )  in  N  is  projected  min- 
imally into  the  hvpersphere  (  '2  )  in  Xn_r.  Obviously,  in  order  that 
the  hypersphere  (-)  should  be  real  the  vertex  of  (  1  )  must  be  imag- 
inary. More  exactly  the  coefficients  <t  ,  c/,,  •  •  •.  <in  ,  mu>t  be  real  and 
'/,;  pun1  imaginary. 

The  coordinates  of  the  vertex  of  a  hypercoue  in  N;  are  then 
essentially  elementary  coi'irdinates  (^  14<i )  of  a  hypersphere  in  \  r 
but  the  radius  of  the  sphere  is  i<tti  instead  of  ^  .  Let  u>,  howevei1, 
introduce  into  Sn  polyspherical  coi'irdinates  bast-d  upon  //  -f  -  hvpei1- 
spheres.  The  coi'irdinates  of  the  vertex  of  a  hypercoue  in  N  and. 
consequently,  of  a  hypersphere  in  \ _j  are  then  //  4-  -  homogeneous 
coordinates  connected  by  a  quadratic  relation.  They  are  therefore 
higher  sphere  coi'irdinates  of  oriented  hvperspheres  in  N  ,.  Hut 
\ve  have  seen  that  the  polvspherical  coi'irdinates  in  X  are  jtrojec- 
tive  eoi'irdinates  of  points  on  a  hypersphere  in  N,,^.  \N*e  have, 
therefore : 


\\'e  havi1  in  this  \vav  obtained  a  geometric  construction  by  \vhieh. 
for  example,  oriented  sphei'es  in  N.  mav  be  brought  into  a  one- 
to-one  relation  xvith  points  on  a  hypersphere  in  N.. 


EXERCISES 

1.    Slm\v   analvtieallv    Ilia1    a    point    .'•   :.'•,:•••:.'•    .,    on    the    li\  )"•!•- 
plieri-  ,/•,-  4-  ./•.:  -)-...-(-  ./•;;  .  ]  T-  0  in  .s';  jirojcrt  s  liv  the  doulile  projci-t  inn 
f   tlii-   text    into   t  he    h  Vpersjiliere  i  /./',    4-  .;•       ,  )  (  £f  +  •  •  •  +  s:T        '  —  '-'•''  v 
-----  L'  ./•„  .  ,  t\  _  o  4-  (  /'.'•„  -  .'•„  .  ,  i  -=  0  in  i,,  _  ,.. 


420  .V- DIMENSIONAL  (JEOMET1IV 

~.    Establish  the  following  relations  between  >'.,  >'  ,  and  >'  ,  <j>  bein 
a  hypersphere  in  >'  : 

>'  >'  >' 

4  3 

A  ]>oiiit  on  $.  A  point. 

A    hyperplane    sec-  A  sphere. 
t  ion  of  (f>. 

A  section  of  $  by  a  A  point  sphere.  A     special     sphere 

tangent   hvperplane.  complex. 

A     niininiuin     line  A  minimum  line.  A  pencil  of  tangent 

on    <j>.  spheres. 

A  ininiinnm  plane  A  niiniinuni  plane  A  bundle  of  tan- 
on  (ft.  of  second  kind.  gent  spheres. 

A   section   of   (f>   liy  A   hvpersurfaee    of  A    sphere    complex 

any  >4'.  order   //.  of   order  >;. 

A    minimum  curve  A  minimum  curve.  A      series      of      -s} 

on    <f>.  spheres, each  of  \vhidi 

is  tanur»'nt   to  1  lie  con- 

>eeut  i\  e    0!le. 

REFERENCES 

>/-/('  »v  >i<-<iini  tr>/  : 

("IHM.I r». i:.  Line  and  S])liorp  Gonmrtry  (>ci'  reference1  at  cii«l  of  l';in  III). 
J.h  >  ;/i  nun  try  : 

IIri>»i«\.  Kuiiiiiii-r's  (^uii'lric  Surfiii'e.      ( 'ainhriili^e  I'niver-ity  1'ivss. 
.1 1  --op,  Tri-;risc  mi  the  Lint1  ( '"iii]i!i-\.    ( 'aint'i'i'l_i'  I'liiver-ity  l'iv-<. 
K'i:vi'.-.  La  L'»'"inet rif  rt;_lt''c  et  "t-s  apitlicatimis.    (iauthii-r-Villars. 
1'i.i'Ki.i:.  Neue  (ieoniftrie  iles  IJ;uiair>.    Tcal'iier. 

I'hicki-r's  w..rk  is  tin-  nrL'inal  anthnrity.  It  i--  fumti'il  ln-ro  fm1  i:s  lii.-tnriral 
v:\lut'.  'I'll''  Muclt-nt  \vill  ]>r«ilial>ly  tiii'l  it  inni-e  cnnveiiiciit  tu  CMn>u!t  thr  nther 
text-,  tli--  scujif  nf  which  is  .-urtiri.'iitl y  iii'licatf'!  l>y  ihcir  titi-s. 

(i.  m, ,.!,-,/  ,,f  h  ilhli-  tixt'ilt*  : 

,Ii  ir  i  i  i:r.T,  ( -I   >;netrii'  a  ipiatre  ilinirn<i"iis.    f ian t lii IT- Vi liars, 
M  \\-\iNi;.  (if"iii'-iry  nf  K"Ui'  I)iii!'-n>i":]<.    'I'ln-  Mannillati  ( 'mnpany. 
Manning's  l»i  •!•;   i-  -•  •  •'  •  ••.-.  Jnuffri't's  analytic.     l-'.-]"-cia!    nirntinn   .-!M"iM    l>p 
f  tin    !  i.-iMj-ical  account  in  ManniiiLf's  int  MHlnciinn,  \viih  cnjiinns  refert-nci  - 
•  .  • ;  .     ;->  ratlin-. 

F"r  !_r'-iirral    (i-i'.iiiicii-iniial    L''-"in»-try   reference   \\il!   }«•   ma^lr   1«   t!if   f'-!;"\\  i::_- 
•  -.  \\    .  -  fiiiiii'l  i->]''-ciall\     -i-t'iil  in  jirenarinu'  hi-  ti-xi  : 

K  i  i  i  \.    "  I  "i-l'f-r    r.iiiii'iii'i1'  iiiH-t  rie      IP!    IIH  • : :  -•  lie    ( .•  >  imrt  rie. "      \liif fn-in>ifi^''tn 
Si.i,t:i..  "  Stn  I  mi  iiiiiin-r'i  'ji;ali;i;Mr.>- 

-i   ,.;."     \t,  •    - .     /.  '/,;    r«il  •   'I'll'    >••;,«;.    .//    '/'-„•;„.,. 

Si          i    >-  :•'•  -.    V,,!.    XXXVI.    1--:.. 

\'i.i:i'M.-i..  "  !>'•!   •  •    M\  i-i-lii-n  N'l-rliiiltni-i-i'1  <i'-r  IJaiiiiM'  vim  vi-r- 

I.-IK-II  Din  ,  •  ,:  ;        |,  ,l;is  I'ri!  ,-ip  iles  rn.ji.-iren.- mnl  SclnicMejis." 

M  ";  •  'i  >'  •  i,<   -!M   ''  •  .  V   :.  XIX.  I--.:. 


INDEX 

(The  numbers  refer  to  pajjes) 

Absolute,  370  Conjugates,  harmonic,  18 

Alliiif  tran.sfuniuitiuii,  102  Contact  traiisformatioiis,  120,  2">s 

Alible,      10"),      HIT,      1SS.     204,     3b'V.     -11.");  Coordinates.    1..'!;    point,    8,    L'7.   31.    138, 

btHween  circles,  144;  beUvt-en. spheres,  KM.  ISO,  l'.»3.  282.  2*s.  3(12,  388  ;  line, 

280.  344  ;  of  parallelism,  112  11,  38,  301.  3o:,.  41o:   plane.   12.  I'.'T  ; 

Asymptotes,  33  circle.  171.  177;   sphere.  341.  343 

Asymptotic  lines,  301  Ciirrclatiuii,  ^8.  ^4ti 

Axis,  of  nuiia1.  H;    of  pencil  of  jilanes.  Cosines,  direction.  H»l,  377 

IL';  of   iniadric,   230;    of  special    line  Cross  ratio.  lt» 

complex,  311  ;   of  any  complex,  318  Curvature,  lines  of,  338 

Curve.  AD.  ;")8,  200 

ISa>e  of  raiiiri-,  8  Cuspidal  ed-e,  212 

liiciivular  curve,  174.281  Cyclic,  174 

IJrianclion's  theorem.  70  Cj'c'liJe,  Dupin's,  274,  350  ;  general.  27'J, 

Handle,  of  planes,   HKi.  1U8;   of  sjiheres,  2'J7 

2»i8,  2U3,  342;    of   lines.  300 ;    of   tan-  Cylindroid,  323 
LienL  spheres,  351 

Deferent.  •_",!',) 

(Vnter.  of  eoiiie,  32  ;  of  quadric.  224.  227  Derive  of  >pace  in  n  dimensic.ins,  3'JO 

( 'haracteri.stic  of  surface.  211  Drsanrues.  theorem  of,  4.") 

Circle,  30;   at  inlinity,  181  Developable  surface.  20S.  214 

Circle  coordinates,  171,   177  Diameter,  of  parabola,  (!7 ;  of  line  com- 

Cirele  points  at  inlinity.  30.  \n~>  1'lex,  31H 

( 'lai rant's  equation,   137  Diameters,  conjugate,  of  conic.  (14  ;   con- 
Class,  of  curve,  ~>~< :   of  surface,  207;  of  jugate,  of  i|uadric.  224 

line  complex  and  congruence,  308  Diametral  plane.  221.  230 

Clifford  parallels.  ~2'>'>  Dilation.  13t» 

Collincat  ions.  72,  2  lo.  2">o.  413  Direction.  1*8;    in  four  dimensions,  3H8  ; 

Complex,  of  cii'eles,  1  I1. i.  1  72.  1  73.  1  7'. * ;  of  in  n  dimensions.  4  1  •"> 

spheres.  2tJ',»,2(,)3. 341.341'.. 3.".3  ;  of  lines,  Distance,  2',».    I3H,  283.  2!M)  ;   jirojective, 

3os.    310,    3Hi.    317.   32S  ;    rosiiiHiilar,  108.   111.   ll.">.   117.  2">4.  2  .").">  :    in  foni' 

332;   ti  t  rahedral.  333  ;   tangent.  3">3  dimensions,     308  :    in     n    ilimeiiMoiis, 

( 'oiifonnal  transformation,  li't;  -111 

Congruence,  of  circles.  172.  17H;  of  lines.  Double  circle  of  conq.lex.  174 

308.  322.  33."..  33(1;   normal  line.  338;  Double  pail'  of  correlation,  l»o.  217 

of  >pheivs.  318  Duality.  2:    point  an<l    line  in  plain-,  lo. 

Conies,  32  ;    pairs  cif.  !i.">  .",i; ;    tetracydieal    plane   and    ijiiadric 

( 'oiijuiiatt;  jioints  and  spaces  in  )i  dimeii-  surface.     1(11.     1»1.",.    2">o  :      ]>oint     and 

sioiis,  31*3  plane.  I'.i'.i ;    line  and  .sphere.  3 .".7  ;    line 

Conjugate  pi.lar  lines,  ,,(  ^  line  complex,  in  t  liree  dimensions  and  point  in  four. 

311;   of  a  i|iiadrie.  223  3s  1.  -1  lo 

121 


422  1XDEX 

F.llip.-oid.  22S  7i-line  and  ji-poim.  44 

F.lliptic  space.  11~>.  2-V>  Non-Euclidean  geometry.  112 

Null  sphere.  1S2 

Focal  carve.  :ii>o  Null  system.  24S.  321 

F<  'cal  point-  of  line  congruence.  33ti,  33S 

Foci  of  conic.  <>•">  Order,  of   plane  curve.  •">:>;    of   surface. 

Form,  algebraic.  140  20.">.  220  :  of  line  complex,  congruence, 

and  series.  3(>8 
< irientation  of  spheres.  :>44 
orthogonal  circles.  14"),  172.  17S 
Orthogonal  spheres.  2ti7.  2btj,  341 

llexasphei ica!  coordinates,  38(3  Osculating  plane.  202 

HomoloL'v.  s"> ;    axis  of.  4','.  70:    center 

of.  4'.!,  7(1  Pappus,  theorem  of.  4S 

Horicycle.  114  Parabolic  space.  117.  L'OO 

Hyperbolic  >pace.  110.  2-V1  Paraboloid.  22',t 

Hypcrboloid.  22S  Paralleli.-m,  2S.  112,  187.  37o.  41ti ;  com- 

Hyperplane.  :-!t'.2.  :)•!'.':    at  inlinity.  3fis.  plete  and  simple.  371  :  Clifford,  2oo 

414:    polar.  :;s!  Pa.-cal's  theorem.  7."> 

Hy]ier-]>here.  .'!7n.  :>.">.  41:;  Pedal  transformation.  l:)l 

II\  per>urface  of  .-econd  order.  :!s2,  :;'.i2        Pencil,  of  point.-.  S;  of  lines.  11.  .''.7.  . '!',». 
liypocycle.  114  ;;nti :  of  ]>lanes.  12.  P.iii;  of  conies.  l>4; 

t>f  circles.   14»i;    of  spheres.  2(i»i,  2'.».';, 

Imaginary  element.  2  ,'!42  :   of  tangent  spheres.  :;.">(! 

Imaginary  line.  ls4.  I'.M  Perpendicularity,  111.   l'.«i,  41»',;     coni- 

linairinary  plane.  K^7  plete   perpendicularity  and   semiper- 

Inlinity.  :J :   locus  at.  s.  2>.  }:}'.>.  142.  Hi").  jiendicularity.  :!7"> 

l^ti.  2M.  :;»;>.  414  Per.-pecti\  ity.  21 

Invariant.  7  Plane.   Is").    l'.»7.  2S">.  :]»12  ;    at  intinity, 

Inversion,  121.  124.  l-'i'i.  2»il.  27n.  L".U  l,si;;   of  lines.  ;;o7 

i         ;  ition.   1">:   of   line  complexes.  :',^~ .        Plane  coiirdinates.  1H7 
411  :    of  sphere  complexes,  o47  Plane  element.  2")'.* 

Planes,  completely  and  .-imply  parallel. 

•oordinates.  Soil  ;)71  ;    completely    perpendicular    and 

Kummi-r'>  surface.  .');J2  seiniperpentlicular.  -\~,~> 

Pliicker  coordinates.  .'J<U 

Line.  •   .  lations  of.  L'7.  :;.">.  P,'">.  P.<7.  :M2.        Pliicker's  com]ilex  surface.  ;;34 
:Jss  :   at   intinity.  2s:    properand   im-        Point,  e.jiiation  of.  :)'.i.  P.<7 
proper.    lv-l:     comjiletely   and  unjoin-        Point-curve  t  ransformat  ion.  1 27.  2'i:!.  .'J(J1 
pli-l'-ly  imaginary.  I'.M  Point-point  tran.-fonnation.  120.  2'JO 

Line  .  ooiilinate-.  lo.  :>.  ;;ol  Point  -]ihere.  1S2.  is").  2S."> 

Lin--  element.  1:;;1  I'oint-surface  transformation.  2''>2 

l.ol   icliev.-kian  Lr'-onieiry.  112  Polar,    \vith    re-pect    to   ]'oii,t    pair.    2ii; 

\\illl    re.-pect   to  cill'Ve  of   -ecolld  order. 
M   ._'      '  .    In}  ."ill;      \\ith    I'e-pect     to    curve    of    Second 

M  '                   me-.   1'.'2  cla.-s.    7n ;    in   treiieral.    11H;    with    re- 

•        es.  :J7s  spect  to  surface  of  second  order.  2±i ; 

^l            mi  lii  •  -.   lsl.  1s'1.  U7s  uith    re-pect    to    -urta<-e    of    .-econd 

n  im  plane-.   lx-\   P.m.  :_'>">.  :17^                    cla.-s.  2.'Js  :    \\ith  re.-pect  to  lineal'  line 


INDKX  4i>:J 

complex,  .')!.'>;    with  respect   tn  quad-  Singular  planes.  I'l'l.  222.  :;2'.» 

ratic  line  complex,  :J2S  ;   with  respect  Singular  points.  ~p2,  "i*.  2<ni.  2'.'»;.  :;L".I.  :>:; 

to  hypersurface,  .'is!,  ;-J!i:J  Space,  linear,  :{hh  :   on  <|ua<lric.   l<il 

1'olar  lines,  conjugate.  22.'!.  :!14  Speciali/ed  ijiiailric.  :','.i~, 

I'olar  spaces,  conjugate.  :;;i:;  Sphere.  2oti.  2*  t  :   oriented.  :;H 

I'ower  of  point,  \\ith  re.-pect   to  circle.  Sphere  coordinates.  341 

I  ."•(  I:    with  respect  to  sphere.  2*7  Spherical  -comet  ry.   lit; 

1'rojection.  I'll;   Mereo^ruphic.  K'.:.'.  4n7.  Splieroimailric.  L'sl 

•11s  ;   minimum.  41H  Surface,   in   point    eonnlinates.  L'II."I;     in 
1'rojective  ueoinetry.  in  jilane.   ItH  :    in  plane  coordinates, -Jl.'i ;   analla^mat;.-. 

three  ilimeiisioiis.  24'J;  on  qiiatlric.  sur-  274.  I".'1.1 :     .-iiiLrnlai'.   :',:',\.   '.',">•'< :     Kuin 

face,  li'iii;   in  ?i  dimensions,  oHM  mer's.  :];!j  :    l'luckci-'>.  :;:M 

1'rojective  mea.-uivincnt,  1U7.  -i'jo 

I'rojectivity,  lo,  20  Tangent  circles.   17,^.  i".'"i 

Pseudo  circle,  ll;;  Taiment  hyperplam  s.  ;>:; 

Taiment  line,  to  ciir\e,  ~>\,  2UO;   to  siu1- 
Quad rankle,  complete.  44  face.  20") 

Quadrilateral,  complete,  44  Tangent  line  complexes.  :L'^ 

Tangent  plane  to  surface.  I'tit; 

/•-flat.  ;!^S  'I'anirent  plane>.  ;J4"i 

Kadical  axis.  20S  Tangent  sphere  complexes.  :).").••} 

Kadical  center.  L'I'.'.I  Taimeiit  spheres.  L".I.",.  ;;(.',.  :;.')() 

Radical  plane.  -jt'>7  Tetracyclieal  coiirdinutes.  l.'JN 

iire.  of  points.  H;   of  conies.  71  Thread.  •_'.">. 


monic.  IS  Transformation,  defined.  4  ;    ahMne,  In2: 

I  etlectioii.  H)4  contact.  l2n.  2">S;   inversion.  121.  lot;. 

I  e-ulu>.  o2»;  2ol.  27n.  2H 1  :   linear.   1:1.   7s.  ss.  ].-,(. 

1  elativity.  111)  111'.'.  2ld.  2ld.  2H1:    metrical,  lol.   1  .V,. 

1  iemannian  geometry.  1K5  24!».  2!'l:  projective.  2H.  K»t.  24U.  2">o  : 

1  im;  surface.  27")  jiedal.      l.'Jl:      I'oint-point.      12(1.    2(i(>; 

lotation.  lu:)  point-curve.   127.  2'I.'!  ;    point-surface, 

1  ulin^s  on  ijuadric.  2:12  202;  quadric  inversion.  121  ;   recipro- 

cal  radius.  124.  2ul.  27n 

Series,  line.  .'JOS.  .'124:   sphere.  .'14'. i  Translation.  Id.'! 
Sheaf  of  planes,  12 

Singular  complex  of  circles.  174 
Singular  lines.  .">4.  (i7.  -']2'.i 


Date  Due 


OCT 


TT7U 


CAT      NO      . -       • 


3    1970   00177   8 


A    001  429985 


